# Further Applications of Trigonometry

## Parametric Equations

Consider the path a moon follows as it orbits a planet, which simultaneously rotates around the sun, as seen in . At any moment, the moon is located at a particular spot relative to the planet. But how do we write and solve the equation for the position of the moon when the distance from the planet, the speed of the moon’s orbit around the planet, and the speed of rotation around the sun are all unknowns? We can solve only for one variable at a time.
In this section, we will consider sets of equations given by
   and
   where
   is the independent variable of time. We can use these parametric equations in a number of applications when we are looking for not only a particular position but also the direction of the movement. As we trace out successive values of
  the orientation of the curve becomes clear. This is one of the primary advantages of using parametric equations: we are able to trace the movement of an object along a path according to time. We begin this section with a look at the basic components of parametric equations and what it means to parameterize a curve. Then we will learn how to eliminate the parameter, translate the equations of a curve defined parametrically into rectangular equations, and find the parametric equations for curves defined by rectangular equations.

### Parameterizing a Curve

When an object moves along a curve—or curvilinear path—in a given direction and in a given amount of time, the position of the object in the plane is given by the x-coordinate and the y-coordinate. However, both
  and

vary over time and so are functions of time. For this reason, we add another variable, the parameter, upon which both
  and
  are dependent functions. In the example in the section opener, the parameter is time,
  The
  position of the moon at time,
  is represented as the function
  and the
  position of the moon at time,
  is represented as the function
  Together,
   and
   are called parametric equations, and generate an ordered pair
  Parametric equations primarily describe motion and direction.
When we parameterize a curve, we are translating a single equation in two variables, such as
  and
  into an equivalent pair of equations in three variables,
  and
  One of the reasons we parameterize a curve is because the parametric equations yield more information: specifically, the direction of the object’s motion over time.
When we graph parametric equations, we can observe the individual behaviors of
  and of
  There are a number of shapes that cannot be represented in the form
  meaning that they are not functions. For example, consider the graph of a circle, given as
  Solving for
  gives
  or two equations:
  and
  If we graph
  and
  together, the graph will not pass the vertical line test, as shown in . Thus, the equation for the graph of a circle is not a function.
However, if we were to graph each equation on its own, each one would pass the vertical line test and therefore would represent a function. In some instances, the concept of breaking up the equation for a circle into two functions is similar to the concept of creating parametric equations, as we use two functions to produce a non-function. This will become clearer as we move forward.

### Eliminating the Parameter

In many cases, we may have a pair of parametric equations but find that it is simpler to draw a curve if the equation involves only two variables, such as
  and
  Eliminating the parameter is a method that may make graphing some curves easier. However, if we are concerned with the mapping of the equation according to time, then it will be necessary to indicate the orientation of the curve as well. There are various methods for eliminating the parameter
  from a set of parametric equations; not every method works for every type of equation. Here we will review the methods for the most common types of equations.

### Eliminating the Parameter from Polynomial, Exponential, and Logarithmic Equations

For polynomial, exponential, or logarithmic equations expressed as two parametric equations, we choose the equation that is most easily manipulated and solve for
  We substitute the resulting expression for

into the second equation. This gives one equation in
  and

### Eliminating the Parameter from Trigonometric Equations

Eliminating the parameter from trigonometric equations is a straightforward substitution. We can use a few of the familiar trigonometric identities and the Pythagorean Theorem.
First, we use the identities:
Solving for
  and
  we have
Then, use the Pythagorean Theorem:
Substituting gives

### Finding Cartesian Equations from Curves Defined Parametrically

When we are given a set of parametric equations and need to find an equivalent Cartesian equation, we are essentially “eliminating the parameter.” However, there are various methods we can use to rewrite a set of parametric equations as a Cartesian equation. The simplest method is to set one equation equal to the parameter, such as
  In this case,
  can be any expression. For example, consider the following pair of equations.
Rewriting this set of parametric equations is a matter of substituting
  for
  Thus, the Cartesian equation is

### Finding Parametric Equations for Curves Defined by Rectangular Equations

Although we have just shown that there is only one way to interpret a set of parametric equations as a rectangular equation, there are multiple ways to interpret a rectangular equation as a set of parametric equations. Any strategy we may use to find the parametric equations is valid if it produces equivalency. In other words, if we choose an expression to represent
  and then substitute it into the
  equation, and it produces the same graph over the same domain as the rectangular equation, then the set of parametric equations is valid. If the domain becomes restricted in the set of parametric equations, and the function does not allow the same values for
  as the domain of the rectangular equation, then the graphs will be different.

### Key Concepts

1. Parameterizing a curve involves translating a rectangular equation in two variables,
  and
  into two equations in three variables, x, y, and t. Often, more information is obtained from a set of parametric equations. See , , and .
2. Sometimes equations are simpler to graph when written in rectangular form. By eliminating
  an equation in
  and
  is the result.
3. To eliminate
  solve one of the equations for
  and substitute the expression into the second equation. See , , , and .
4. Finding the rectangular equation for a curve defined parametrically is basically the same as eliminating the parameter. Solve for
  in one of the equations, and substitute the expression into the second equation. See .
5. There are an infinite number of ways to choose a set of parametric equations for a curve defined as a rectangular equation.
6. Find an expression for
  such that the domain of the set of parametric equations remains the same as the original rectangular equation. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, eliminate the parameter
  to rewrite the parametric equation as a Cartesian equation.
For the following exercises, rewrite the parametric equation as a Cartesian equation by building an
   table.
For the following exercises, parameterize (write parametric equations for) each Cartesian equation by setting
   or by setting

For the following exercises, parameterize (write parametric equations for) each Cartesian equation by using
   and
  Identify the curve.

### Technology

For the following exercises, use the table feature in the graphing calculator to determine whether the graphs intersect.
For the following exercises, use a graphing calculator to complete the table of values for each set of parametric equations.

### Extensions


# Further Applications of Trigonometry

## Parametric Equations: Graphs

While not every fan (or team manager) appreciates it, baseball and many other sports have become dependent on analytics, which involve complex data recording and quantitative evaluation used to understand and predict behavior. The earliest influence of analytics was mostly statistical; more recently, physics and other sciences have come into play. Foremost among these is the focus on launch angle and exit velocity, which when at certain values can almost guarantee a home run. On the other hand, emphasis on launch angle and focusing on home runs rather than overall hitting results in far more outs. Consider the following situation: it is the bottom of the ninth inning, with two outs and two players on base. The home team is losing by two runs. The batter swings and hits the baseball at 140 feet per second and at an angle of approximately
  to the horizontal. How far will the ball travel? Will it clear the fence for a game-winning home run? The outcome may depend partly on other factors (for example, the wind), but mathematicians can model the path of a projectile and predict approximately how far it will travel using parametric equations. In this section, we’ll discuss parametric equations and some common applications, such as projectile motion problems.

### Graphing Parametric Equations by Plotting Points

In lieu of a graphing calculator or a computer graphing program, plotting points to represent the graph of an equation is the standard method. As long as we are careful in calculating the values, point-plotting is highly dependable.

### Applications of Parametric Equations

Many of the advantages of parametric equations become obvious when applied to solving real-world problems. Although rectangular equations in x and y give an overall picture of an object's path, they do not reveal the position of an object at a specific time. Parametric equations, however, illustrate how the values of x and y change depending on t, as the location of a moving object at a particular time.
A common application of parametric equations is solving problems involving projectile motion. In this type of motion, an object is propelled forward in an upward direction forming an angle of
            to the horizontal, with an initial speed of
           and at a height
            above the horizontal.
The path of an object propelled at an inclination of
              to the horizontal, with initial speed
  and at a height
              above the horizontal, is given by
where
             accounts for the effects of gravity and
              is the initial height of the object. Depending on the units involved in the problem, use
  or
  The equation for
             gives horizontal distance, and the equation for
              gives the vertical distance.

### Key Concepts

1. When there is a third variable, a third parameter on which
           and
           depend, parametric equations can be used.
2. To graph parametric equations by plotting points, make a table with three columns labeled
           and
           Choose values for
           in increasing order. Plot the last two columns for
           and
           See  and .
3. When graphing a parametric curve by plotting points, note the associated t-values and show arrows on the graph indicating the orientation of the curve. See  and .
4. Parametric equations allow the direction or the orientation of the curve to be shown on the graph. Equations that are not functions can be graphed and used in many applications involving motion. See .
5. Projectile motion depends on two parametric equations:
  and
  Initial velocity is symbolized as
   represents the initial angle of the object when thrown, and
            represents the height at which the object is propelled.

### Section Exercises

### Verbal

### Graphical

For the following exercises, graph each set of parametric equations by making a table of values. Include the orientation on the graph.
For the following exercises, sketch the curve and include the orientation.
For the following exercises, graph the equation and include the orientation. Then, write the Cartesian equation.
For the following exercises, graph the equation and include the orientation.
For the following exercises, use the parametric equations for integers a and b:
For the following exercises, describe the graph of the set of parametric equations.
For the following exercises, use a graphing utility to graph on the window
     by
     on the domain
     for the following values of
     and
    , and include the orientation.

### Technology

For the following exercises, look at the graphs that were created by parametric equations of the form
  Use the parametric mode on the graphing calculator to find the values of
              and
              to achieve each graph.
For the following exercises, use a graphing utility to graph the given parametric equations.

### Extensions

For the following exercises, use this scenario: A dart is thrown upward with an initial velocity of 64 ft/s at an angle of elevation of 52°. Consider the position of the dart at any time
             Neglect air resistance.
For the following exercises, look at the graphs of each of the four parametric equations. Although they look unusual and beautiful, they are so common that they have names, as indicated in each exercise. Use a graphing utility to graph each on the indicated domain.


# Further Applications of Trigonometry

## Vectors

An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140°. A north wind (from north to south) is blowing at 16.2 miles per hour, as shown in . What are the ground speed and actual bearing of the plane?
Ground speed refers to the speed of a plane relative to the ground. Airspeed refers to the speed a plane can travel relative to its surrounding air mass. These two quantities are not the same because of the effect of wind. In an earlier section, we used triangles to solve a similar problem involving the movement of boats. Later in this section, we will find the airplane’s groundspeed and bearing, while investigating another approach to problems of this type. First, however, let’s examine the basics of vectors.

### A Geometric View of Vectors

A vector is a specific quantity drawn as a line segment with an arrowhead at one end. It has an initial point, where it begins, and a terminal point, where it ends. A vector is defined by its magnitude, or the length of the line, and its direction, indicated by an arrowhead at the terminal point. Thus, a vector is a directed line segment. There are various symbols that distinguish vectors from other quantities:
1. Lower case, boldfaced type, with or without an arrow on top such as

2. Given initial point
  and terminal point
  a vector can be represented as
  The arrowhead on top is what indicates that it is not just a line, but a directed line segment.
3. Given an initial point of
  and terminal point
  a vector may be represented as

This last symbol
   has special significance. It is called the standard position. The position vector has an initial point
  and a terminal point
  To change any vector into the position vector, we think about the change in the x-coordinates and the change in the y-coordinates. Thus, if the initial point of a vector
  is
  and the terminal point is
  then the position vector is found by calculating
In , we see the original vector
  and the position vector

### Finding Magnitude and Direction

To work with a vector, we need to be able to find its magnitude and its direction. We find its magnitude using the Pythagorean Theorem or the distance formula, and we find its direction using the inverse tangent function.

### Performing Vector Addition and Scalar Multiplication

Now that we understand the properties of vectors, we can perform operations involving them. While it is convenient to think of the vector
  as an arrow or directed line segment from the origin to the point
  vectors can be situated anywhere in the plane. The sum of two vectors  and , or vector addition, produces a third vector + , the resultant vector.
To find  + , we first draw the vector , and from the terminal end of , we drawn the vector . In other words, we have the initial point of  meet the terminal end of . This position corresponds to the notion that we move along the first vector and then, from its terminal point, we move along the second vector. The sum  +  is the resultant vector because it results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning of  to the end of  in a straight path, as shown in .
Vector subtraction is similar to vector addition. To find  − , view it as  + (−). Adding − is reversing direction of  and adding it to the end of  . The new vector begins at the start of   and stops at the end point of −. See  for a visual that compares vector addition and vector subtraction using parallelograms.

### Multiplying By a Scalar

While adding and subtracting vectors gives us a new vector with a different magnitude and direction, the process of multiplying a vector by a scalar, a constant, changes only the magnitude of the vector or the length of the line. Scalar multiplication has no effect on the direction unless the scalar is negative, in which case the direction of the resulting vector is opposite the direction of the original vector.

### Finding Component Form

In some applications involving vectors, it is helpful for us to be able to break a vector down into its components. Vectors are comprised of two components: the horizontal component is the
  direction, and the vertical component is the
  direction. For example, we can see in the graph in  that the position vector
  comes from adding the vectors 1 and 2. We have 1 with initial point
  and terminal point

We also have 2 with initial point
  and terminal point

Therefore, the position vector is
Using the Pythagorean Theorem, the magnitude of 1 is 2, and the magnitude of 2 is 3. To find the magnitude of , use the formula with the position vector.
The magnitude of  is
  To find the direction, we use the tangent function

Thus, the magnitude of is
  and the direction is
  off the horizontal.

### Finding the Unit Vector in the Direction of v

In addition to finding a vector’s components, it is also useful in solving problems to find a vector in the same direction as the given vector, but of magnitude 1. We call a vector with a magnitude of 1 a unit vector. We can then preserve the direction of the original vector while simplifying calculations.
Unit vectors are defined in terms of components. The horizontal unit vector is written as
  and is directed along the positive horizontal axis. The vertical unit vector is written as
  and is directed along the positive vertical axis. See .

### Performing Operations with Vectors in Terms of i and j

So far, we have investigated the basics of vectors: magnitude and direction, vector addition and subtraction, scalar multiplication, the components of vectors, and the representation of vectors geometrically. Now that we are familiar with the general strategies used in working with vectors, we will represent vectors in rectangular coordinates in terms of  and .

### Performing Operations on Vectors in Terms of i and j

When vectors are written in terms of and we can carry out addition, subtraction, and scalar multiplication by performing operations on corresponding components.

### Calculating the Component Form of a Vector: Direction

We have seen how to draw vectors according to their initial and terminal points and how to find the position vector. We have also examined notation for vectors drawn specifically in the Cartesian coordinate plane using
  For any of these vectors, we can calculate the magnitude. Now, we want to combine the key points, and look further at the ideas of magnitude and direction.
Calculating direction follows the same straightforward process we used for polar coordinates. We find the direction of the vector by finding the angle to the horizontal. We do this by using the basic trigonometric identities, but with replacing

### Finding the Dot Product of Two Vectors

As we discussed earlier in the section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the dot product and the cross product. We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses.
The dot product of two vectors involves multiplying two vectors together, and the result is a scalar.

### Key Concepts

1. The position vector has its initial point at the origin. See .
2. If the position vector is the same for two vectors, they are equal. See .
3. Vectors are defined by their magnitude and direction. See .
4. If two vectors have the same magnitude and direction, they are equal. See .
5. Vector addition and subtraction result in a new vector found by adding or subtracting corresponding elements. See .
6. Scalar multiplication is multiplying a vector by a constant. Only the magnitude changes; the direction stays the same. See  and .
7. Vectors are comprised of two components: the horizontal component along the positive x-axis, and the vertical component along the positive y-axis. See .
8. The unit vector in the same direction of any nonzero vector is found by dividing the vector by its magnitude.
9. The magnitude of a vector in the rectangular coordinate system is
  See .
10. In the rectangular coordinate system, unit vectors may be represented in terms of   and   where represents the horizontal component and represents the vertical component. Then,  = a + b  is a scalar multiple of by real numbers
  See  and .
11. Adding and subtracting vectors in terms of i and j consists of adding or subtracting corresponding coefficients of i and corresponding coefficients of j. See .
12.  A vector v = a + b is written in terms of magnitude and direction as
  See .
13. The dot product of two vectors is the product of the terms plus the product of the terms. See .
14. We can use the dot product to find the angle between two vectors.  and .
15. Dot products are useful for many types of physics applications. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the two vectors and are equal, where has an initial point
  and a terminal point
  and   has an initial point
  and a terminal point
 .
For the following exercises, use the vectors  =  + 5,  = −2− 3,  and  = 4 − .
For the following exercises, use the given vectors to compute  + ,  − , and 2 − 3.
For the following exercises, find a unit vector in the same direction as the given vector.
For the following exercises, find the magnitude and direction of the vector,

### Graphical

For the following exercises, given draw 3 and

For the following exercises, use the vectors shown to sketch  + ,  − , and 2.

For the following exercises, use the vectors shown to sketch 2 + .
For the following exercises, use the vectors shown to sketch  − 3.
For the following exercises, write the vector shown in component form.

### Extensions

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

### Real-World Applications

### Chapter Review Exercises

### Non-right Triangles: Law of Sines

For the following exercises, assume
  is opposite side
  is opposite side
  and
  is opposite side
  Solve each triangle, if possible. Round each answer to the nearest tenth.

### Non-right Triangles: Law of Cosines

### Polar Coordinates

For the following exercises, convert the given Cartesian equation to a polar equation.
For the following exercises, convert the given polar equation to a Cartesian equation.
For the following exercises, convert to rectangular form and graph.

### Polar Coordinates: Graphs

For the following exercises, test each equation for symmetry.

### Polar Form of Complex Numbers

For the following exercises, find the absolute value of each complex number.
Write the complex number in polar form.
For the following exercises, convert the complex number from polar to rectangular form.
For the following exercises, find the product
  in polar form.
For the following exercises, find the quotient
  in polar form.
For the following exercises, find the powers of each complex number in polar form.
For the following exercises, evaluate each root.
For the following exercises, plot the complex number in the complex plane.

### Parametric Equations

For the following exercises, eliminate the parameter
  to rewrite the parametric equation as a Cartesian equation.

### Parametric Equations: Graphs

For the following exercises, make a table of values for each set of parametric equations, graph the equations, and include an orientation; then write the Cartesian equation.

### Vectors

For the following exercises, determine whether the two vectors, and are equal, where has an initial point
  and a terminal point
  and has an initial point
  and a terminal point

For the following exercises, use the vectors and to evaluate the expression.
For the following exercises, find a unit vector in the same direction as the given vector.
For the following exercises, find the magnitude and direction of the vector.
For the following exercises, calculate

### Practice Test

Given
  and
  evaluate each expression.
For the following exercises, use the vectors  =  − 3 and  = 2 + 3.


# Systems of Equations and Inequalities

## Introduction to Systems of Equations and Inequalities

At the start of the Second World War, British military and intelligence officers recognized that defeating Nazi Germany would require the Allies to know what the enemy was planning. This task was complicated by the fact that the German military transmitted all of its communications through a presumably uncrackable code created by a machine called Enigma. The Germans had been encoding their messages with this machine since the early 1930s, and were so confident in its security that they used it for everyday military communications as well as highly important strategic messages. Concerned about the increasing military threat, other European nations began working to decipher the Enigma codes. Poland was the first country to make significant advances when it trained and recruited a new group of codebreakers: math students from Poznań University. With the help of intelligence obtained by French spies, Polish mathematicians, led by Marian Rejewski, were able to decipher initial codes and later to understand the wiring of the machines; eventually they create replicas. However, the German military eventually increased the complexity of the machines by adding additional rotors, requiring a new method of decryption.
The machine attached letters on a keyboard to three, four, or five rotors (depending on the version), each with 26 starting positions that could be set prior to encoding; a decryption code (called a cipher key) essentially conveyed these settings to the message recipient, and allowed people to interpret the message using another Enigma machine. Even with the simpler three-rotor scrambler, there were 17,576 different combinations of starting positions (26 x 26 x 26); plus the machine had numerous other methods of introducing variation. Not long after the war started, the British recruited a team of brilliant codebreakers to crack the Enigma code. The codebreakers, led by Alan Turing, used what they knew about the Enigma machine to build a mechanical computer that could crack the code. And that knowledge of what the Germans were planning proved to be a key part of the ultimate Allied victory of Nazi Germany in 1945.
The Enigma is perhaps the most famous cryptographic device ever known. It stands as an example of the pivotal role cryptography has played in society. Now, technology has moved cryptanalysis to the digital world.
Many ciphers are designed using invertible matrices as the method of message transference, as finding the inverse of a matrix is generally part of the process of decoding. In addition to knowing the matrix and its inverse, the receiver must also know the key that, when used with the matrix inverse, will allow the message to be read.
In this chapter, we will investigate matrices and their inverses, and various ways to use matrices to solve systems of equations. First, however, we will study systems of equations on their own: linear and nonlinear, and then partial fractions. We will not be breaking any secret codes here, but we will lay the foundation for future courses.


# Systems of Equations and Inequalities

## Systems of Linear Equations: Two Variables

### Learning Objectives

1. Determine whether an ordered pair is a solution of a system of equations (IA 4.1.1)
2. Solve a system of linear equations by graphing (IA 4.1.2)

### Objective: Determine whether an ordered pair is a solution of a system of equations (IA 4.1.1)

A system of linear equations is a group of two or more linear equations.  For example,

is a system of linear equations

A solution to a system of linear equations is an ordered pair x,y that is a solution to every equation in the system.

### Practice Makes Perfect

Determine whether the ordered pairs are solutions to the given system.

 at  and

### Solve a system of linear equations by graphing (IA 4.1.2)

### Practice Makes Perfect

A skateboard manufacturer introduces a new line of boards. The manufacturer tracks its costs, which is the amount it spends to produce the boards, and its revenue, which is the amount it earns through sales of its boards. How can the company determine if it is making a profit with its new line? How many skateboards must be produced and sold before a profit is possible? In this section, we will consider linear equations with two variables to answer these and similar questions.

### Introduction to Systems of Equations

In order to investigate situations such as that of the skateboard manufacturer, we need to recognize that we are dealing with more than one variable and likely more than one equation. A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all equations in the system at the same time. Some linear systems may not have a solution and others may have an infinite number of solutions. In order for a linear system to have a unique solution, there must be at least as many equations as there are variables. Even so, this does not guarantee a unique solution.
In this section, we will look at systems of linear equations in two variables, which consist of two equations that contain two different variables. For example, consider the following system of linear equations in two variables.
The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. In this example, the ordered pair (4, 7) is the solution to the system of linear equations. We can verify the solution by substituting the values into each equation to see if the ordered pair satisfies both equations. Shortly we will investigate methods of finding such a solution if it exists.
In addition to considering the number of equations and variables, we can categorize systems of linear equations by the number of solutions. A consistent system of equations has at least one solution. A consistent system is considered to be an independent system if it has a single solution, such as the example we just explored. The two lines have different slopes and intersect at one point in the plane. A consistent system is considered to be a dependent system if the equations have the same slope and the same y-intercepts. In other words, the lines coincide so the equations represent the same line. Every point on the line represents a coordinate pair that satisfies the system. Thus, there are an infinite number of solutions.
Another type of system of linear equations is an inconsistent system, which is one in which the equations represent two parallel lines. The lines have the same slope and different y-intercepts. There are no points common to both lines; hence, there is no solution to the system.

###  Solving Systems of Equations by Graphing

There are multiple methods of solving systems of linear equations. For a system of linear equations in two variables, we can determine both the type of system and the solution by graphing the system of equations on the same set of axes.

### Solving Systems of Equations by Substitution

Solving a linear system in two variables by graphing works well when the solution consists of integer values, but if our solution contains decimals or fractions, it is not the most precise method. We will consider two more methods of solving a system of linear equations that are more precise than graphing. One such method is solving a system of equations by the substitution method, in which we solve one of the equations for one variable and then substitute the result into the second equation to solve for the second variable. Recall that we can solve for only one variable at a time, which is the reason the substitution method is both valuable and practical.

### Solving Systems of Equations in Two Variables by the Addition Method

A third method of solving systems of linear equations is the addition method. In this method, we add two terms with the same variable, but opposite coefficients, so that the sum is zero. Of course, not all systems are set up with the two terms of one variable having opposite coefficients. Often we must adjust one or both of the equations by multiplication so that one variable will be eliminated by addition.

### Identifying Inconsistent Systems of Equations Containing Two Variables

Now that we have several methods for solving systems of equations, we can use the methods to identify inconsistent systems. Recall that an inconsistent system consists of parallel lines that have the same slope but different
  -intercepts. They will never intersect. When searching for a solution to an inconsistent system, we will come up with a false statement, such as

### Expressing the Solution of a System of Dependent Equations Containing Two Variables

Recall that a dependent system of equations in two variables is a system in which the two equations represent the same line. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line. After using substitution or addition, the resulting equation will be an identity, such as

###  Using Systems of Equations to Investigate Profits

Using what we have learned about systems of equations, we can return to the skateboard manufacturing problem at the beginning of the section. The skateboard manufacturer’s revenue function is the function used to calculate the amount of money that comes into the business. It can be represented by the equation
  where
  quantity and
  price. The revenue function is shown in orange in .
The cost function is the function used to calculate the costs of doing business. It includes fixed costs, such as rent and salaries, and variable costs, such as utilities. The cost function is shown in blue in . The
  -axis represents quantity in hundreds of units. The y-axis represents either cost or revenue in hundreds of dollars.
The point at which the two lines intersect is called the break-even point. We can see from the graph that if 700 units are produced, the cost is $3,300 and the revenue is also $3,300. In other words, the company breaks even if they produce and sell 700 units. They neither make money nor lose money.
The shaded region to the right of the break-even point represents quantities for which the company makes a profit. The shaded region to the left represents quantities for which the company suffers a loss. The profit function is the revenue function minus the cost function, written as
  Clearly, knowing the quantity for which the cost equals the revenue is of great importance to businesses.

### Key Concepts

1. A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously.
2. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. See .
3. Systems of equations are classified as independent with one solution, dependent with an infinite number of solutions, or inconsistent with no solution.
4. One method of solving a system of linear equations in two variables is by graphing. In this method, we graph the equations on the same set of axes. See .
5. Another method of solving a system of linear equations is by substitution. In this method, we solve for one variable in one equation and substitute the result into the second equation. See .
6. A third method of solving a system of linear equations is by addition, in which we can eliminate a variable by adding opposite coefficients of corresponding variables. See .
7. It is often necessary to multiply one or both equations by a constant to facilitate elimination of a variable when adding the two equations together. See , , and .
8. Either method of solving a system of equations results in a false statement for inconsistent systems because they are made up of parallel lines that never intersect. See .
9. The solution to a system of dependent equations will always be true because both equations describe the same line. See .
10. Systems of equations can be used to solve real-world problems that involve more than one variable, such as those relating to revenue, cost, and profit. See  and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the given ordered pair is a solution to the system of equations.
For the following exercises, solve each system by substitution.
For the following exercises, solve each system by addition.
For the following exercises, solve each system by any method.

### Graphical

For the following exercises, graph the system of equations and state whether the system is consistent, inconsistent, or dependent and whether the system has one solution, no solution, or infinite solutions.

### Technology

For the following exercises, use the intersect function on a graphing device to solve each system. Round all answers to the nearest hundredth.

### Extensions

For the following exercises, solve each system in terms of
  and
  where
  are nonzero numbers. Note that
  and

### Real-World Applications

For the following exercises, solve for the desired quantity.
For the following exercises, use a system of linear equations with two variables and two equations to solve.


# Systems of Equations and Inequalities

## Systems of Linear Equations: Three Variables

### Learning Objectives

1. Determine whether an ordered triple is a solution of a system of three linear equations with three variables (IA 4.4.1)

2. Solve a system of three linear equations with three variables (IA 4.4.2)

### Objective 1: Determine whether an ordered triple is a solution of a system of three linear equations with three variables (IA 4.4.1)

A linear equation with three variables where a, b, c, and d are real numbers and a, b, and c are not all 0, is of the form  . The graph of a linear equation with three variables is a plane.

A system of linear equations with three variables is a set of linear equations with three variables. For example,

is a system of linear equations with three variables.
Solutions of a system of equations are the values of the variables that make all the equations true. A solution is represented by an ordered triple (x,y,z).

### Practice Makes Perfect

Determine whether the ordered pairs are solutions to the given system.

### Objective 2: Solve a system of three linear equations with three variables (IA 4.4.2)

When we solve a system of linear equations with three variables, we have many possible solutions.

The solutions are summarized in the table below.

### Practice Makes Perfect

Jordi received an inheritance of $12,000 that he divided into three parts and invested in three ways: in a money-market fund paying 3% annual interest; in municipal bonds paying 4% annual interest; and in mutual funds paying 7% annual interest. Jordi invested $4,000 more in mutual funds than in municipal bonds. He earned $670 in interest the first year. How much did Jordi invest in each type of fund?
Understanding the correct approach to setting up problems such as this one makes finding a solution a matter of following a pattern. We will solve this and similar problems involving three equations and three variables in this section. Doing so uses similar techniques as those used to solve systems of two equations in two variables. However, finding solutions to systems of three equations requires a bit more organization and a touch of visualization.

### Solving Systems of Three Equations in Three Variables

In order to solve systems of equations in three variables, known as three-by-three systems, the primary tool we will be using is called Gaussian elimination, named after the prolific German mathematician Karl Friedrich Gauss. While there is no definitive order in which operations are to be performed, there are specific guidelines as to what type of moves can be made. We may number the equations to keep track of the steps we apply. The goal is to eliminate one variable at a time to achieve upper triangular form, the ideal form for a three-by-three system because it allows for straightforward back-substitution to find a solution
  which we call an ordered triple. A system in upper triangular form looks like the following:
The third equation can be solved for
  and then we back-substitute to find

 and

 To write the system in upper triangular form, we can perform the following operations:
1. Interchange the order of any two equations.
2. Multiply both sides of an equation by a nonzero constant.
3. Add a nonzero multiple of one equation to another equation.
The solution set to a three-by-three system is an ordered triple

 Graphically, the ordered triple defines the point that is the intersection of three planes in space. You can visualize such an intersection by imagining any corner in a rectangular room.  A corner is defined by three planes: two adjoining walls and the floor (or ceiling). Any point where two walls and the floor meet represents the intersection of three planes.

### Identifying Inconsistent Systems of Equations Containing Three Variables

Just as with systems of equations in two variables, we may come across an inconsistent system of equations in three variables, which means that it does not have a solution that satisfies all three equations. The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. The process of elimination will result in a false statement, such as

 or some other contradiction.

### Expressing the Solution of a System of Dependent Equations Containing Three Variables

We know from working with systems of equations in two variables that a dependent system of equations has an infinite number of solutions. The same is true for dependent systems of equations in three variables. An infinite number of solutions can result from several situations. The three planes could be the same, so that a solution to one equation will be the solution to the other two equations. All three equations could be different but they intersect on a line, which has infinite solutions. Or two of the equations could be the same and intersect the third on a line.

### Key Concepts

1. A solution set is an ordered triple

 that represents the intersection of three planes in space. See .
2. A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated.   The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero multiple of one equation to another equation. See .
3. Systems of three equations in three variables are useful for solving many different types of real-world problems. See .
4. A system of equations in three variables is inconsistent if no solution exists. After performing elimination operations, the result is a contradiction. See .
5. Systems of equations in three variables that are inconsistent could result from three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location.
6. A system of equations in three variables is dependent if it has an infinite number of solutions. After performing elimination operations, the result is an identity. See .
7. Systems of equations in three variables that are dependent could result from three identical planes, three planes intersecting at a line, or two identical planes that intersect the third on a line.

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the ordered triple given is the solution to the system of equations.
For the following exercises, solve each system by elimination.
For the following exercises, solve each system by Gaussian elimination.

### Extensions

For the following exercises, solve the system for
  and

### Real-World Applications


# Systems of Equations and Inequalities

## Systems of Nonlinear Equations and Inequalities: Two Variables

### Learning Objectives

1. Graph a parabola (IA 11.2.1)
2. Graph a circle (IA 11.1.4)

### Objective 1: Graph a parabola (IA 11.2.1)

A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.

### Practice Makes Perfect

### Objective 2: Graph a circle (IA 11.1.4)

Any equation of the form  is the standard form of the equation of a circle with center, (h,k) and radius. We can then graph the circle on a rectangular coordinate system using the center and radius.

### Practice Makes Perfect

Graph a circle.
Halley’s Comet () orbits the sun about once every 75 years. Its path can be considered to be a very elongated ellipse. Other comets follow similar paths in space. These orbital paths can be studied using systems of equations. These systems, however, are different from the ones we considered in the previous section because the equations are not linear.
In this section, we will consider the intersection of a parabola and a line, a circle and a line, and a circle and an ellipse. The methods for solving systems of nonlinear equations are similar to those for linear equations.

### Solving a System of Nonlinear Equations Using Substitution

A system of nonlinear equations is a system of two or more equations in two or more variables containing at least one equation that is not linear. Recall that a linear equation can take the form
 Any equation that cannot be written in this form in nonlinear. The substitution method we used for linear systems is the same method we will use for nonlinear systems. We solve one equation for one variable and then substitute the result into the second equation to solve for another variable, and so on. There is, however, a variation in the possible outcomes.

###  Intersection of a Parabola and a Line

There are three possible types of solutions for a system of nonlinear equations involving a parabola and a line.

### Intersection of a Circle and a Line

Just as with a parabola and a line, there are three possible outcomes when solving a system of equations representing a circle and a line.

### Solving a System of Nonlinear Equations Using Elimination

We have seen that substitution is often the preferred method when a system of equations includes a linear equation and a nonlinear equation. However, when both equations in the system have like variables of the second degree, solving them using elimination by addition is often easier than substitution. Generally, elimination is a far simpler method when the system involves only two equations in two variables (a two-by-two system), rather than a three-by-three system, as there are fewer steps. As an example, we will investigate the possible types of solutions when solving a system of equations representing a circle and an ellipse.

### Graphing a Nonlinear Inequality

All of the equations in the systems that we have encountered so far have involved equalities, but we may also encounter systems that involve inequalities. We have already learned to graph linear inequalities by graphing the corresponding equation, and then shading the region represented by the inequality symbol. Now, we will follow similar steps to graph a nonlinear inequality so that we can learn to solve systems of nonlinear inequalities. A nonlinear inequality is an inequality containing a nonlinear expression. Graphing a nonlinear inequality is much like graphing a linear inequality.
Recall that when the inequality is greater than,
 or less than,
 the graph is drawn with a dashed line. When the inequality is greater than or equal to,
 or less than or equal to,
 the graph is drawn with a solid line. The graphs will create regions in the plane, and we will test each region for a solution. If one point in the region works, the whole region works. That is the region we shade. See .

### Graphing a System of Nonlinear Inequalities

Now that we have learned to graph nonlinear inequalities, we can learn how to graph systems of nonlinear inequalities. A system of nonlinear inequalities is a system of two or more inequalities in two or more variables containing at least one inequality that is not linear. Graphing a system of nonlinear inequalities is similar to graphing a system of linear inequalities. The difference is that our graph may result in more shaded regions that represent a solution than we find in a system of linear inequalities. The solution to a nonlinear system of inequalities is the region of the graph where the shaded regions of the graph of each inequality overlap, or where the regions intersect, called the feasible region.

### Key Concepts

1. There are three possible types of solutions to a system of equations representing a line and a parabola: (1) no solution, the line does not intersect the parabola; (2) one solution, the line is tangent to the parabola; and (3) two solutions, the line intersects the parabola in two points. See .
2. There are three possible types of solutions to a system of equations representing a circle and a line: (1) no solution, the line does not intersect the circle; (2) one solution, the line is tangent to the circle; (3) two solutions, the line intersects the circle in two points. See .
3. There are five possible types of solutions to the system of nonlinear equations representing an ellipse and a circle: (1) no solution, the circle and the ellipse do not intersect; (2) one solution, the circle and the ellipse are tangent to each other; (3) two solutions, the circle and the ellipse intersect in two points; (4) three solutions, the circle and ellipse intersect in three places; (5) four solutions, the circle and the ellipse intersect in four points. See .
4. An inequality is graphed in much the same way as an equation, except for > or <, we draw a dashed line and shade the region containing the solution set. See .
5. Inequalities are solved the same way as equalities, but solutions to systems of inequalities must satisfy both inequalities. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, solve the system of nonlinear equations using substitution.
For the following exercises, solve the system of nonlinear equations using elimination.
For the following exercises, use any method to solve the system of nonlinear equations.
For the following exercises, use any method to solve the nonlinear system.

### Graphical

For the following exercises, graph the inequality.
For the following exercises, graph the system of inequalities. Label all points of intersection.

### Extensions

For the following exercises, graph the inequality.
For the following exercises, find the solutions to the nonlinear equations with two variables.

### Technology

For the following exercises, solve the system of inequalities. Use a calculator to graph the system to confirm the answer.

### Real-World Applications

For the following exercises, construct a system of nonlinear equations to describe the given behavior, then solve for the requested solutions.


# Systems of Equations and Inequalities

## Partial Fractions

### Learning Objectives

1. Find the least common denominator of rational expressions (IA 7.2.3)
2. Solve a system of equations by elimination (IA 4.1.4)

### Objective 1: Find the least common denominator of rational expressions (IA 7.2.3)

A rational expression is an expression of the form  where p and q are polynomials and  .
  are examples of rational expressions.

### Practice Makes Perfect

Find the least common denominator of the following rationals:

### Objective 2: Solve a system of equations by elimination (IA 4.1.4)

### Partial Fraction Decomposition

When we add rational expressions with unlike denominators such as  and , we first need to find the LCD, then rewrite each fraction with the common denominator, and finally add the two numerators.

We want to do the opposite now.

Given a rational expression like,  we would like to rewrite it as an addition of two simpler rational expressions  and  . Our goal is to find the values of A and B such that

Earlier in this chapter, we studied systems of two equations in two variables, systems of three equations in three variables, and nonlinear systems. Here we introduce another way that systems of equations can be utilized—the decomposition of rational expressions.
Fractions can be complicated; adding a variable in the denominator makes them even more so. The methods studied in this section will help simplify the concept of a rational expression.

### Decomposing   Where Q(x) Has Only Nonrepeated Linear Factors

Recall the algebra regarding adding and subtracting rational expressions. These operations depend on finding a common denominator so that we can write the sum or difference as a single, simplified rational expression. In this section, we will look at partial fraction decomposition, which is the undoing of the procedure to add or subtract rational expressions. In other words, it is a return from the single simplified rational expression to the original expressions, called the partial fraction.
For example, suppose we add the following fractions:
We would first need to find a common denominator,

Next, we would write each expression with this common denominator and find the sum of the terms.
Partial fraction decomposition is the reverse of this procedure. We would start with the solution and rewrite (decompose) it as the sum of two fractions.
We will investigate rational expressions with linear factors and quadratic factors in the denominator where the degree of the numerator is less than the degree of the denominator. Regardless of the type of expression we are decomposing, the first and most important thing to do is factor the denominator.
When the denominator of the simplified expression contains distinct linear factors, it is likely that each of the original rational expressions, which were added or subtracted, had one of the linear factors as the denominator. In other words, using the example above, the factors of
 are
 the denominators of the decomposed rational expression. So we will rewrite the simplified form as the sum of individual fractions and use a variable for each numerator. Then, we will solve for each numerator using one of several methods available for partial fraction decomposition.

### Decomposing    Where Q(x) Has Repeated Linear Factors

Some fractions we may come across are special cases that we can decompose into partial fractions with repeated linear factors. We must remember that we account for repeated factors by writing each factor in increasing powers.

### Decomposing     Where Q(x) Has a Nonrepeated Irreducible Quadratic Factor

So far, we have performed partial fraction decomposition with expressions that have had linear factors in the denominator, and we applied numerators
   or
   representing constants. Now we will look at an example where one of the factors in the denominator is a quadratic expression that does not factor. This is referred to as an irreducible quadratic factor. In cases like this, we use a linear numerator such as
   etc.

### Decomposing      When Q(x) Has a Repeated Irreducible Quadratic Factor

Now that we can decompose a simplified rational expression with an irreducible quadratic factor, we will learn how to do partial fraction decomposition when the simplified rational expression has repeated irreducible quadratic factors. The decomposition will consist of partial fractions with linear numerators over each irreducible quadratic factor represented in increasing powers.

### Key Concepts

1. Decompose

 by writing the partial fractions as

 Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a system of equations. See .
2. The decomposition of

 with repeated linear factors must account for the factors of the denominator in increasing powers. See .
3. The decomposition of

 with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor, as in

 See .
4. In the decomposition of

 where

 has a repeated irreducible quadratic factor, when the irreducible quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers as

See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the decomposition of the partial fraction for the nonrepeating linear factors.
For the following exercises, find the decomposition of the partial fraction for the repeating linear factors.
For the following exercises, find the decomposition of the partial fraction for the irreducible nonrepeating quadratic factor.
For the following exercises, find the decomposition of the partial fraction for the irreducible repeating quadratic factor.

### Extensions

For the following exercises, find the partial fraction expansion.
For the following exercises, perform the operation and then find the partial fraction decomposition.


# Systems of Equations and Inequalities

## Matrices and Matrix Operations

### Learning Objectives

1. Write the augmented matrix for a system of equations (IA 4.5.1)
2. Add, subtract matrices and multiply a matrix by a scalar

### Objective 1: Write the augmented matrix for a system of equations (IA 4.5.1)

A matrix is a rectangular array of numbers arranged in rows and columns.
A matrix with m rows and n columns has dimension m×n.

Each number in the matrix is called an element or entry in the matrix.

The matrix on the left below has 2 rows and 3 columns and so it has order 2×3. We say it is a 2 by 3 matrix.

We will use a matrix to represent systems of equations.

Each column then would be the coefficients of one of the variables in the system or the constants.

A vertical line replaces the equal signs.

We call the resulting matrix the augmented matrix for the system of equations.

### Practice Makes Perfect

Write each system of linear equations as an augmented matrix

### Objective 2: Add, subtract matrices and multiply a matrix by a scalar

We add or subtract matrices by adding or subtracting corresponding entries.

In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions. We can add or subtract a 3 × 3 matrix and another 3 × 3 matrix, but we cannot add or subtract a 2 × 3 matrix and a 3 × 3 matrix because some entries in one matrix will not have a corresponding entry in the other matrix.

The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication.

### Practice Makes Perfect

Perform the indicated operations

Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season.  shows the needs of both teams.
A goal costs $300; a ball costs $10; and a jersey costs $30. How can we find the total cost for the equipment needed for each team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and used for calculating other information. Then, we will be able to calculate the cost of the equipment.

### Finding the Sum and Difference of Two Matrices

To solve a problem like the one described for the soccer teams, we can use a matrix, which is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Each number is an entry, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [  ] or (  ), and are usually named with capital letters. For example, three matrices named
 and
  are shown below.

###  Describing Matrices

A matrix is often referred to by its size or dimensions:
  indicating
     rows and
     columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix
     identified as
     we look for the entry in row
     column
     In matrix
     shown below, the entry in row 2, column 3 is

A square matrix is a matrix with dimensions
  meaning that it has the same number of rows as columns. The
     matrix above is an example of a square matrix.
A row matrix is a matrix consisting of one row with dimensions

A column matrix is a matrix consisting of one column with dimensions

A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic matrix operations.

###  Adding and Subtracting Matrices

We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.
In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions. We can add or subtract a
  matrix and another
  matrix, but we cannot add or subtract a
  matrix and a
  matrix because some entries in one matrix will not have a corresponding entry in the other matrix.

### Finding Scalar Multiples of a Matrix

Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. Recall that a scalar is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication.
Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment. They estimate that 15% more equipment is needed in both labs. The school’s current inventory is displayed in .
Converting the data to a matrix, we have
To calculate how much computer equipment will be needed, we multiply all entries in matrix
       by 0.15.
We must round up to the next integer, so the amount of new equipment needed is
Adding the two matrices as shown below, we see the new inventory amounts.
This means
Thus, Lab A will have 18 computers, 19 computer tables, and 19 chairs; Lab B will have 32 computers, 40 computer tables, and 40 chairs.

### Finding the Product of Two Matrices

In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If
       is an
  matrix and
       is an
  matrix, then the product matrix
       is an
  matrix. For example, the product
       is possible because the number of columns in
       is the same as the number of rows in
       If the inner dimensions do not match, the product is not defined.
We multiply entries of
       with entries of
       according to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers.
To obtain the entries in row
       of
       we multiply the entries in row
       of
       by column
       in
       and add. For example, given matrices
       and
       where the dimensions of
       are
       and the dimensions of
       are
       the product of
       will be a
       matrix.
Multiply and add as follows to obtain the first entry of the product matrix

1. To obtain the entry in row 1, column 1 of
         multiply the first row in
         by the first column in
         and add.

2. To obtain the entry in row 1, column 2 of
         multiply the first row of
         by the second column in
         and add.

3. To obtain the entry in row 1, column 3 of
         multiply the first row of
         by the third column in
         and add.

We proceed the same way to obtain the second row of
       In other words, row 2 of
       times column 1 of
       row 2 of
       times column 2 of
       row 2 of
       times column 3 of
       When complete, the product matrix will be

### Key Concepts

1. A matrix is a rectangular array of numbers. Entries are arranged in rows and columns.
2. The dimensions of a matrix refer to the number of rows and the number of columns. A
  matrix has three rows and two columns. See .
3. We add and subtract matrices of equal dimensions by adding and subtracting corresponding entries of each matrix. See , , , and .
4. Scalar multiplication involves multiplying each entry in a matrix by a constant. See .
5. Scalar multiplication is often required before addition or subtraction can occur. See .
6. Multiplying matrices is possible when inner dimensions are the same—the number of columns in the first matrix must match the number of rows in the second.
7. The product of two matrices,
  and
  is obtained by multiplying each entry in row 1 of
  by each entry in column 1 of
  then multiply each entry of row 1 of
  by each entry in columns 2 of
  and so on. See  and .
8. Many real-world problems can often be solved using matrices. See .
9. We can use a calculator to perform matrix operations after saving each matrix as a matrix variable. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.
For the following exercises, use the matrices below to perform scalar multiplication.
For the following exercises, use the matrices below to perform matrix multiplication.
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint:

)
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint:

)

### Technology

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.

### Extensions

For the following exercises, use the matrix below to perform the indicated operation on the given matrix.


# Systems of Equations and Inequalities

## Solving Systems with Gaussian Elimination

### Learning Objectives

1. Use row operations on a matrix (IA 4.5.2)
2. Solve systems of equations using matrices (IA 4.5.3)

### Objective 1: Use row operations on a matrix (IA 4.5.2)

In the last section, we learned how to write the augmented matrix for a system of equations.

Once a system of equations is in its augmented matrix form, we will solve by elimination by performing operations on the rows that will lead us to the solution. Our goal will be to get 1 on the diagonal of the matrix and all entries below the diagonal must be zeros.

### Row Operations

In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.

1. Interchange any two rows.
2. Multiply a row by any real number except 0.
3. Add a nonzero multiple of one row to another row.
These actions are called row operations and will help us use the matrix to solve a system of equations.

### Practice Makes Perfect

### Objective 2: Solve systems of equations using matrices (IA 4.5.3)

To solve a system of equations using matrices, we transform the augmented matrix into a matrix in row-echelon form using row operations. For a consistent and independent system of equations, the augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.

Once we get the augmented matrix into row-echelon form, we can write the equivalent system of equations and solve for at least one variable. We then substitute this value in another equation to continue to solve for the other variables.

### Practice Makes Perfect

Carl Friedrich Gauss lived during the late 18th century and early 19th century, but he is still considered one of the most prolific mathematicians in history. His contributions to the science of mathematics and physics span fields such as algebra, number theory, analysis, differential geometry, astronomy, and optics, among others. His discoveries regarding matrix theory changed the way mathematicians have worked for the last two centuries.
We first encountered Gaussian elimination in Systems of Linear Equations: Two Variables. In this section, we will revisit this technique for solving systems, this time using matrices.

### Writing the Augmented Matrix of a System of Equations

A matrix can serve as a device for representing and solving a system of equations. To express a system in matrix form, we extract the coefficients of the variables and the constants, and these become the entries of the matrix. We use a vertical line to separate the coefficient entries from the constants, essentially replacing the equal signs. When a system is written in this form, we call it an augmented matrix.
For example, consider the following
  system of equations.
We can write this system as an augmented matrix:
We can also write a matrix containing just the coefficients. This is called the coefficient matrix.
A three-by-three system of equations such as
has a coefficient matrix
and is represented by the augmented matrix
Notice that the matrix is written so that the variables line up in their own columns: x-terms go in the first column, y-terms in the second column, and z-terms in the third column. It is very important that each equation is written in standard form
  so that the variables line up. When there is a missing variable term in an equation, the coefficient is 0.

### Writing a System of Equations from an Augmented Matrix

We can use augmented matrices to help us solve systems of equations because they simplify operations when the systems are not encumbered by the variables. However, it is important to understand how to move back and forth between formats in order to make finding solutions smoother and more intuitive. Here, we will use the information in an augmented matrix to write the system of equations in standard form.

### Performing Row Operations on a Matrix

Now that we can write systems of equations in augmented matrix form, we will examine the various row operations that can be performed on a matrix, such as addition, multiplication by a constant, and interchanging rows.
Performing row operations on a matrix is the method we use for solving a system of equations. In order to solve the system of equations, we want to convert the matrix to row-echelon form, in which there are ones down the main diagonal from the upper left corner to the lower right corner, and zeros in every position below the main diagonal as shown.
We use row operations corresponding to equation operations to obtain a new matrix that is row-equivalent in a simpler form. Here are the guidelines to obtaining row-echelon form.
1. In any nonzero row, the first nonzero number is a 1. It is called a leading 1.
2. Any all-zero rows are placed at the bottom on the matrix.
3. Any leading 1 is below and to the right of a previous leading 1.
4. Any column containing a leading 1 has zeros in all other positions in the column.
To solve a system of equations we can perform the following row operations to convert the coefficient matrix to row-echelon form and do back-substitution to find the solution.
1. Interchange rows. (Notation:
  )
2. Multiply a row by a constant. (Notation:
  )
3. Add the product of a row multiplied by a constant to another row. (Notation:

Each of the row operations corresponds to the operations we have already learned to solve systems of equations in three variables. With these operations, there are some key moves that will quickly achieve the goal of writing a matrix in row-echelon form. To obtain a matrix in row-echelon form for finding solutions, we use Gaussian elimination, a method that uses row operations to obtain a 1 as the first entry so that row 1 can be used to convert the remaining rows.

### Solving a System of Linear Equations Using Matrices

We have seen how to write a system of equations with an augmented matrix, and then how to use row operations and back-substitution to obtain row-echelon form. Now, we will take row-echelon form a step farther to solve a 3 by 3 system of linear equations. The general idea is to eliminate all but one variable using row operations and then back-substitute to solve for the other variables.

### Key Concepts

1. An augmented matrix is one that contains the coefficients and constants of a system of equations. See .
2. A matrix augmented with the constant column can be represented as the original system of equations. See .
3. Row operations include multiplying a row by a constant, adding one row to another row, and interchanging rows.
4. We can use Gaussian elimination to solve a system of equations. See , , and .
5. Row operations are performed on matrices to obtain row-echelon form. See .
6. To solve a system of equations, write it in augmented matrix form. Perform row operations to obtain row-echelon form. Back-substitute to find the solutions. See  and .
7. A calculator can be used to solve systems of equations using matrices. See .
8. Many real-world problems can be solved using augmented matrices. See  and .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, write the augmented matrix for the linear system.
For the following exercises, write the linear system from the augmented matrix.
For the following exercises, solve the system by Gaussian elimination.

### Extensions

For the following exercises, use Gaussian elimination to solve the system.

### Real-World Applications

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution.


# Systems of Equations and Inequalities

## Solving Systems with Inverses

### Learning Objectives

1. Evaluate the determinant of a 2×2 matrix (IA 4.6.1)
2. Evaluate the determinant of a 3x3 matrix (IA 4.6.2)

### Objective 1: Evaluate the determinant of a 2×2 matrix (IA 4.6.1)

If a matrix has the same number of rows and columns, we call it a square matrix. Each square matrix has a real number associated with it called its determinant.

### Practice Makes Perfect

Find the determinant of the 2x2 matrices.

### Objective 2: Evaluate the determinant of a 3×3 matrix (IA 4.6.2)

To evaluate the determinant of a 3×3 matrix, we must be able to evaluate the minor of an entry in the determinant.

The minor of an entry is the 2×2 determinant found by eliminating the row and column in the 3×3 determinant that contains the entry.

For example, to find the minor of entry a1, we eliminate the row and column which contain it. So, we eliminate the first row and first column. Then we write the 2×2 determinant that remains.

To find the minor of entry b2, we eliminate the row and column that contain it. So, we eliminate the second row and second column. Then we write the 2×2 determinant that remains.

### Strategy for evaluating the determinant of a 3x3 matrix

To evaluate a 3×3 determinant we can expand by minors using any row or column. Choosing a row or column other than the first row sometimes makes the work easier.

When we expand by any row or column, we must be careful about the sign of the terms in the expansion. To determine the sign of the terms, we use the following sign pattern chart.

### Expanding by minors along the first row to evaluate a 3x3 determinant.

To evaluate a 3×3 determinant by expanding by minors along the first row, we use the following pattern:

NOTE: We can evaluate the determinant of a matrix by expanding minors along any row or column. When a row or a column has a zero entry, expanding by that row or column results in less calculations.

### Practice Makes Perfect

Soriya plans to invest $10,500 into two different bonds to spread out her risk. The first bond has an annual return of 10%, and the second bond has an annual return of 6%. In order to receive an 8.5% return from the two bonds, how much should Soriya invest in each bond? What is the best method to solve this problem?
There are several ways we can solve this problem. As we have seen in previous sections, systems of equations and matrices are useful in solving real-world problems involving finance. After studying this section, we will have the tools to solve the bond problem using the inverse of a matrix.

### Finding the Inverse of a Matrix

We know that the multiplicative inverse of a real number
       is
       and
       For example,
       and
       The multiplicative inverse of a matrix is similar in concept, except that the product of matrix
       and its inverse
       equals the identity matrix. The identity matrix is a square matrix containing ones down the main diagonal and zeros everywhere else. We identify identity matrices by
       where
       represents the dimension of the matrix. Observe the following equations.
The identity matrix acts as a 1 in matrix algebra. For example,

A matrix that has a multiplicative inverse has the properties
A matrix that has a multiplicative inverse is called an invertible matrix. Only a square matrix may have a multiplicative inverse, as the reversibility,
       is a requirement. Not all square matrices have an inverse, but if
       is invertible, then
       is unique. We will look at two methods for finding the inverse of a
       matrix and a third method that can be used on both
       and
  matrices.

### Finding the Multiplicative Inverse Using Matrix Multiplication

We can now determine whether two matrices are inverses, but how would we find the inverse of a given matrix? Since we know that the product of a matrix and its inverse is the identity matrix, we can find the inverse of a matrix by setting up an equation using matrix multiplication.

### Finding the Multiplicative Inverse by Augmenting with the Identity

Another way to find the multiplicative inverse is by augmenting with the identity. When matrix
         is transformed into
         the augmented matrix
         transforms into

For example, given
augment
         with the identity
Perform row operations with the goal of turning
         into the identity.
1. Switch row 1 and row 2.

2. Multiply row 2 by
  and add to row 1.

3. Multiply row 1 by
  and add to row 2.

4. Add row 2 to row 1.

5. Multiply row 2 by

The matrix we have found is

### Finding the Multiplicative Inverse of 2×2 Matrices Using a Formula

When we need to find the multiplicative inverse of a
         matrix, we can use a special formula instead of using matrix multiplication or augmenting with the identity.
If
         is a
         matrix, such as
the multiplicative inverse of
         is given by the formula
where
         If
         then
         has no inverse.

### Finding the Multiplicative Inverse of 3×3 Matrices

Unfortunately, we do not have a formula similar to the one for a
         matrix to find the inverse of a
         matrix. Instead, we will augment the original matrix with the identity matrix and use row operations to obtain the inverse.
Given a

        matrix
augment
         with the identity matrix
To begin, we write the augmented matrix with the identity on the right and
         on the left. Performing elementary row operations so that the identity matrix appears on the left, we will obtain the inverse matrix on the right. We will find the inverse of this matrix in the next example.

### Solving a System of Linear Equations Using the Inverse of a Matrix

Solving a system of linear equations using the inverse of a matrix requires the definition of two new matrices:
       is the matrix representing the variables of the system, and
       is the matrix representing the constants. Using matrix multiplication, we may define a system of equations with the same number of equations as variables as
To solve a system of linear equations using an inverse matrix, let
       be the coefficient matrix, let
       be the variable matrix, and let
       be the constant matrix. Thus, we want to solve a system
       For example, look at the following system of equations.
From this system, the coefficient matrix is
The variable matrix is
And the constant matrix is

    Then
       looks like
Recall the discussion earlier in this section regarding multiplying a real number by its inverse,
  To solve a single linear equation
       for
       we would simply multiply both sides of the equation by the multiplicative inverse (reciprocal) of
       Thus,
The only difference between a solving a linear equation and a system of equations written in matrix form is that finding the inverse of a matrix is more complicated, and matrix multiplication is a longer process. However, the goal is the same—to isolate the variable.
We will investigate this idea in detail, but it is helpful to begin with a
  system and then move on to a
  system.

### Key Equations

### Key Concepts

1. An identity matrix has the property
       See .
2. An invertible matrix has the property
       See .
3. Use matrix multiplication and the identity to find the inverse of a
       matrix. See .
4. The multiplicative inverse can be found using a formula.  See .
5. Another method of finding the inverse is by augmenting with the identity. See .
6. We can augment a
       matrix with the identity on the right and use row operations to turn the original matrix into the identity, and the matrix on the right becomes the inverse. See .
7. 	Write the system of equations as
       and multiply both sides by the inverse of
       See  and .
8. We can also use a calculator to solve a system of equations with matrix inverses. See .

### Section Exercises

### Verbal

### Algebraic

In the following exercises, show that matrix
         is the inverse of matrix

For the following exercises, find the multiplicative inverse of each matrix, if it exists.
For the following exercises, solve the system using the inverse of a
  matrix.
For the following exercises, solve a system using the inverse of a

        matrix.

### Technology

For the following exercises, use a calculator to solve the system of equations with matrix inverses.

### Extensions

For the following exercises, find the inverse of the given matrix.

### Real-World Applications

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix.


# Systems of Equations and Inequalities

## Solving Systems with Cramer's Rule

### Learning Objectives

1. Use Cramer’s Rule to solve systems of equations (IA 4.6.3)

### Objective 1: Use Cramer’s Rule to solve systems of equations (IA 4.6.3)

Cramer’s Rule uses determinants to solve systems of equations.

### Practice Makes Perfect

### Practice Makes Perfect

We have learned how to solve systems of equations in two variables and three variables, and by multiple methods: substitution, addition, Gaussian elimination, using the inverse of a matrix, and graphing. Some of these methods are easier to apply than others and are more appropriate in certain situations. In this section, we will study two more strategies for solving systems of equations.

### Evaluating the Determinant of a 2×2 Matrix

A determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a square matrix to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an invertible matrix and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.

### Using Cramer’s Rule to Solve a System of Two Equations in Two Variables

We will now introduce a final method for solving systems of equations that uses determinants. Known as Cramer’s Rule, this technique dates back to the middle of the 18th century and is named for its innovator, the Swiss mathematician Gabriel Cramer (1704-1752), who introduced it in 1750 in Introduction à l'Analyse des lignes Courbes algébriques. Cramer’s Rule is a viable and efficient method for finding solutions to systems with an arbitrary number of unknowns, provided that we have the same number of equations as unknowns.
Cramer’s Rule will give us the unique solution to a system of equations, if it exists. However, if the system has no solution or an infinite number of solutions, this will be indicated by a determinant of zero. To find out if the system is inconsistent or dependent, another method, such as elimination, will have to be used.
To understand Cramer’s Rule, let’s look closely at how we solve systems of linear equations using basic row operations. Consider a system of two equations in two variables.
We eliminate one variable using row operations and solve for the other. Say that we wish to solve for
  If equation (2) is multiplied by the opposite of the coefficient of
  in equation (1), equation (1) is multiplied by the coefficient of
  in equation (2), and we add the two equations, the variable
  will be eliminated.
Now, solve for

Similarly, to solve for
  we will eliminate

Solving for
  gives
Notice that the denominator for both
  and
  is the determinant of the coefficient matrix.
We can use these formulas to solve for
  and
  but Cramer’s Rule also introduces new notation:
1.
  determinant of the coefficient matrix
2.
  determinant of the numerator in the solution of

3.
  determinant of the numerator in the solution of

The key to Cramer’s Rule is replacing the variable column of interest with the constant column and calculating the determinants. We can then express
  and
  as a quotient of two determinants.

### Evaluating the Determinant of a 3 × 3 Matrix

Finding the determinant of a 2×2 matrix is straightforward, but finding the determinant of a 3×3 matrix is more complicated. One method is to augment the 3×3 matrix with a repetition of the first two columns, giving a 3×5 matrix. Then we calculate the sum of the products of entries down each of the three diagonals (upper left to lower right), and subtract the products of entries up each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.
Find the determinant of the 3×3 matrix.
1. Augment
  with the first two columns.

2. From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.
3. From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.
The algebra is as follows:

### Using Cramer’s Rule to Solve a System of Three Equations in Three Variables

Now that we can find the determinant of a 3 × 3 matrix, we can apply Cramer’s Rule to solve a system of three equations in three variables. Cramer’s Rule is straightforward, following a pattern consistent with Cramer’s Rule for 2 × 2 matrices. As the order of the matrix increases to 3 × 3, however, there are many more calculations required.
When we calculate the determinant to be zero, Cramer’s Rule gives no indication as to whether the system has no solution or an infinite number of solutions. To find out, we have to perform elimination on the system.
Consider a 3 × 3 system of equations.
where
If we are writing the determinant
  we replace the
  column with the constant column. If we are writing the determinant
  we replace the
  column with the constant column. If we are writing the determinant
  we replace the
  column with the constant column. Always check the answer.

### Understanding Properties of Determinants

There are many properties of determinants. Listed here are some properties that may be helpful in calculating the determinant of a matrix.

### Key Concepts

1. The determinant for
  is
  See .
2. Cramer’s Rule replaces a variable column with the constant column. Solutions are
  See .
3. To find the determinant of a 3×3 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right). See .
4. To solve a system of three equations in three variables using Cramer’s Rule, replace a variable column with the constant column for each desired solution:
  See .
5. Cramer’s Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions. See  and .
6. Certain properties of determinants are useful for solving problems. For example:

### Section Exercises

### Verbal

### Algebraic

For the following exercises, find the determinant.
For the following exercises, solve the system of linear equations using Cramer’s Rule.
For the following exercises, solve the system of linear equations using Cramer’s Rule.

### Technology

For the following exercises, use the determinant function on a graphing utility.

### Real-World Applications

For the following exercises, create a system of linear equations to describe the behavior. Then, calculate the determinant. Will there be a unique solution? If so, find the unique solution.
For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer’s Rule.
For the following exercises, use this scenario: A health-conscious company decides to make a trail mix out of almonds, dried cranberries, and chocolate-covered cashews. The nutritional information for these items is shown in .

### Review Exercises

### Systems of Linear Equations: Two Variables

For the following exercises, determine whether the ordered pair is a solution to the system of equations.
For the following exercises, use substitution to solve the system of equations.
For the following exercises, use addition to solve the system of equations.
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

### Systems of Linear Equations: Three Variables

For the following exercises, solve the system of three equations using substitution or addition.
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

### Systems of Nonlinear Equations and Inequalities: Two Variables

For the following exercises, solve the system of nonlinear equations.
For the following exercises, graph the inequality.
For the following exercises, graph the system of inequalities.

### Partial Fractions

For the following exercises, decompose into partial fractions.

### Matrices and Matrix Operations

For the following exercises, perform the requested operations on the given matrices.

### Solving Systems with Gaussian Elimination

For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.
For the following exercises, write the augmented matrix from the system of linear equations.
For the following exercises, solve the system of linear equations using Gaussian elimination.

### Solving Systems with Inverses

For the following exercises, find the inverse of the matrix.
For the following exercises, find the solutions by computing the inverse of the matrix.
For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

### Solving Systems with Cramer's Rule

For the following exercises, find the determinant.
For the following exercises, use Cramer’s Rule to solve the linear systems of equations.

### Practice Test

Is the following ordered pair a solution to the system of equations?
For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.
For the following exercises, graph the following inequalities.
For the following exercises, write the partial fraction decomposition.
For the following exercises, perform the given matrix operations.
For the following exercises, use Gaussian elimination to solve the systems of equations.
For the following exercises, use the inverse of a matrix to solve the systems of equations.
For the following exercises, use Cramer’s Rule to solve the systems of equations.
For the following exercises, solve using a system of linear equations.


# Analytic Geometry

## Introduction to Analytic Geometry

The Greek mathematician Menaechmus (c. 380–c. 320 BCE) is generally credited with discovering the shapes formed by the intersection of a plane and a right circular cone. Depending on how he tilted the plane when it intersected the cone, he formed different shapes at the intersection–beautiful shapes with near-perfect symmetry.
It was also said that Aristotle may have had an intuitive understanding of these shapes, as he observed the orbit of the planet to be circular. He presumed that the planets moved in circular orbits around Earth, and for nearly 2000 years this was the commonly held belief.
It was not until the Renaissance movement that Johannes Kepler noticed that the orbits of the planet were not circular in nature. His published law of planetary motion in the 1600s changed our view of the solar system forever. He claimed that the sun was at one end of the orbits, and the planets revolved around the sun in an oval-shaped path.
Other objects in the solar system (and perhaps other systems) follow a similar elliptical path, including the spectacular rings of Saturn. Using this understanding as a basis, 19th century mathematicians like James Clerk Maxwell and Sofya Kovalevskaya showed that despite their appearance through the telescopes of the day (and even in current telescopes), the rings are not solid and continuous, but are rather composed of small particles. Even after the Voyager and Cassini missions have provided close-up and detailed data regarding the ring structures, full understanding of their construction relies heavily on mathematical analysis. Of particular interest are the influences of Saturn's moons and moonlets, and the ways they both disrupt and preserve the ring structure.
In this chapter, we will investigate the two-dimensional figures that are formed when a right circular cone is intersected by a plane. We will begin by studying each of three figures created in this manner. We will develop defining equations for each figure and then learn how to use these equations to solve a variety of problems.