# 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.


# Analytic Geometry

## The Ellipse

### Learning Objectives

1. Complete the square of a binomial expression. (IA 9.2.1)
2. Graph a circle. (IA 11.1.4)

### Objective 1: Complete the square of a binomial expression. (IA 9.2.1)

But what happens if we have to solve an equation where the trinomial is not a perfect square?
For example,  ? For these types of equations, we can use a process called completing the square.
Recall  .
We can use the Binomial Squares Pattern to make a perfect square.

### Practice Makes Perfect

Determine what number would have to be added to the given terms to create a perfect square trinomial. Then rewrite as a binomial squared.

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

A circle is all points in a plane that are a fixed distance from a given point in the plane. The given point is called the center, (h, k) and the fixed distance is called the radius, r, of the circle.

The standard or graphing form of the equation of a circle with center, (h, k) and radius, r, is  .
The general form of the equation of a circle is  . If we are given an equation in general form, we can change it to standard, also called the graphing form, by completing the squares in both x  and y. Then we can graph the circle using its center and radius.

### Practice Makes Perfect

Can you imagine standing at one end of a large room and still being able to hear a whisper from a person standing at the other end? The National Statuary Hall in Washington, D.C., shown in , is such a room.Architect of the Capitol. http://www.aoc.gov. Accessed April 15, 2014. It is an semi-circular room called a whispering chamber because the shape makes it possible for sound to travel along the walls and dome. In this section, we will investigate the shape of this room and its real-world applications, including how far apart two people in Statuary Hall can stand and still hear each other whisper.

### Writing Equations of Ellipses in Standard Form

A conic section, or conic, is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines the shape, as shown in .
Conic sections can also be described by a set of points in the coordinate plane. Later in this chapter, we will see that the graph of any quadratic equation in two variables is a conic section. The signs of the equations and the coefficients of the variable terms determine the shape. This section focuses on the four variations of the standard form of the equation for the ellipse. An ellipse is the set of all points
  in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string. Place the thumbtacks in the cardboard to form the foci of the ellipse. Cut a piece of string longer than the distance between the two thumbtacks (the length of the string represents the constant in the definition). Tack each end of the string to the cardboard, and trace a curve with a pencil held taut against the string. The result is an ellipse. See .
Every ellipse has two axes of symmetry. The longer axis is called the major axis, and the shorter axis is called the minor axis. Each endpoint of the major axis is the vertex of the ellipse (plural: vertices), and each endpoint of the minor axis is a co-vertex of the ellipse. The center of an ellipse is the midpoint of both the major and minor axes. The axes are perpendicular at the center. The foci always lie on the major axis, and the sum of the distances from the foci to any point on the ellipse (the constant sum) is greater than the distance between the foci. See .
In this section, we restrict ellipses to those that are positioned vertically or horizontally in the coordinate plane. That is, the axes will either lie on or be parallel to the x- and y-axes. Later in the chapter, we will see ellipses that are rotated in the coordinate plane.
To work with horizontal and vertical ellipses in the coordinate plane, we consider two cases: those that are centered at the origin and those that are centered at a point other than the origin. First we will learn to derive the equations of ellipses, and then we will learn how to write the equations of ellipses in standard form. Later we will use what we learn to draw the graphs.

### Deriving the Equation of an Ellipse Centered at the Origin

To derive the equation of an ellipse centered at the origin, we begin with the foci
  and
  The ellipse is the set of all points
  such that the sum of the distances from
  to the foci is constant, as shown in .
If
  is a vertex of the ellipse, the distance from
  to
  is
  The distance from
  to
  is
  . The sum of the distances from the foci to the vertex is
If
  is a point on the ellipse, then we can define the following variables:
By the definition of an ellipse,
  is constant for any point
  on the ellipse. We know that the sum of these distances is
  for the vertex
  It follows that
  for any point on the ellipse. We will begin the derivation by applying the distance formula. The rest of the derivation is algebraic.
Thus, the standard equation of an ellipse is
  This equation defines an ellipse centered at the origin. If
  the ellipse is stretched further in the horizontal direction, and if
  the ellipse is stretched further in the vertical direction.

### Writing Equations of Ellipses Centered at the Origin in Standard Form

Standard forms of equations tell us about key features of graphs. Take a moment to recall some of the standard forms of equations we’ve worked with in the past: linear, quadratic, cubic, exponential, logarithmic, and so on. By learning to interpret standard forms of equations, we are bridging the relationship between algebraic and geometric representations of mathematical phenomena.
The key features of the ellipse are its center, vertices, co-vertices, foci, and lengths and positions of the major and minor axes. Just as with other equations, we can identify all of these features just by looking at the standard form of the equation. There are four variations of the standard form of the ellipse. These variations are categorized first by the location of the center (the origin or not the origin), and then by the position (horizontal or vertical). Each is presented along with a description of how the parts of the equation relate to the graph. Interpreting these parts allows us to form a mental picture of the ellipse.

### Writing Equations of Ellipses Not Centered at the Origin

Like the graphs of other equations, the graph of an ellipse can be translated. If an ellipse is translated
  units horizontally and
  units vertically, the center of the ellipse will be
  This translation results in the standard form of the equation we saw previously, with
  replaced by
  and y replaced by

###  Graphing Ellipses Centered at the Origin

Just as we can write the equation for an ellipse given its graph, we can graph an ellipse given its equation. To graph ellipses centered at the origin, we use the standard form
  for horizontal ellipses and
  for vertical ellipses.

### Graphing Ellipses Not Centered at the Origin

When an ellipse is not centered at the origin, we can still use the standard forms to find the key features of the graph. When the ellipse is centered at some point,
    we use the standard forms
    for horizontal ellipses and
    for vertical ellipses. From these standard equations, we can easily determine the center, vertices, co-vertices, foci, and positions of the major and minor axes.

### Solving Applied Problems Involving Ellipses

Many real-world situations can be represented by ellipses, including orbits of planets, satellites, moons and comets, and shapes of boat keels, rudders, and some airplane wings. A medical device called a lithotripter uses elliptical reflectors to break up kidney stones by generating sound waves. Some buildings, called whispering chambers, are designed with elliptical domes so that a person whispering at one focus can easily be heard by someone standing at the other focus. This occurs because of the acoustic properties of an ellipse. When a sound wave originates at one focus of a whispering chamber, the sound wave will be reflected off the elliptical dome and back to the other focus. See . In the whisper chamber at the Museum of Science and Industry in Chicago, two people standing at the foci—about 43 feet apart—can hear each other whisper.  When these chambers are placed in unexpected places, such as the ones inside Bush International Airport in Houston and Grand Central Terminal in New York City, they can induce surprised reactions among travelers.

### Key Equations

### Key Concepts

1. An ellipse is the set of all points
   in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
2. When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. See  and .
3. When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. See  and .
4. When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse. See  and .
5. Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.
For the following exercises, find the foci for the given ellipses.

### Graphical

For the following exercises, graph the given ellipses, noting center, vertices, and foci.
For the following exercises, use the given information about the graph of each ellipse to determine its equation.
For the following exercises, given the graph of the ellipse, determine its equation.

###  Extensions

For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula

###  Real-World Applications


# Analytic Geometry

## The Hyperbola

### Learning Objectives

1. Use the Distance Formula. (IA 11.1.1)
2. Graph a hyperbola with center at (0,0). (IA 11.4.1)

### Objective 1: Use the Distance Formula. (IA 11.1.1)

### Practice Makes Perfect

Use the Distance Formula.

### Objective 2: Graph a hyperbola with center at (0,0). (IA 11.4.1)

A hyperbola is all points in a plane where the difference of their distances from two fixed points is constant. Each of the fixed points is called a focus of the hyperbola.
The line through the foci is called the transverse axis. The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola. The midpoint of the segment joining the foci is called the center of the hyperbola. The line perpendicular to the transverse axis that passes through the center is called the conjugate axis. Each piece of the graph is called a branch of the hyperbola.
Notice that, unlike the equation of an ellipse, the denominator of  is not always  and the denominator of  is not always  .
Notice that when the  term is positive, the transverse axis is on the x-axis. When the  term is positive, the transverse axis is on the y-axis.

### Practice Makes Perfect

 Graph a hyperbola with center at (0,0).

What do paths of comets, supersonic booms, ancient Grecian pillars, and natural draft cooling towers have in common? They can all be modeled by the same type of conic. For instance, when something moves faster than the speed of sound, a shock wave in the form of a cone is created. A portion of a conic is formed when the wave intersects the ground, resulting in a sonic boom. See .
Most people are familiar with the sonic boom created by supersonic aircraft, but humans were breaking the sound barrier long before the first supersonic flight. The crack of a whip occurs because the tip is exceeding the speed of sound. The bullets shot from many firearms also break the sound barrier, although the bang of the gun usually supersedes the sound of the sonic boom.

### Locating the Vertices and Foci  of a Hyperbola

In analytic geometry, a hyperbola is a conic section formed by intersecting a right circular cone with a plane at an angle such that both halves of the cone are intersected. This intersection produces two separate unbounded curves that are mirror images of each other. See .
Like the ellipse, the hyperbola can also be defined as a set of points in the coordinate plane. A hyperbola is the set of all points
  in a plane such that the difference of the distances between
  and the foci is a positive constant.
Notice that the definition of a hyperbola is very similar to that of an ellipse. The distinction is that the hyperbola is defined in terms of the difference of two distances, whereas the ellipse is defined in terms of the sum of two distances.
As with the ellipse, every hyperbola has two axes of symmetry. The transverse axis is a line segment that passes through the center of the hyperbola and has vertices as its endpoints. The foci lie on the line that contains the transverse axis. The conjugate axis is perpendicular to the transverse axis and has the co-vertices as its endpoints. The center of a hyperbola is the midpoint of both the transverse and conjugate axes, where they intersect. Every hyperbola also has two asymptotes that pass through its center. As a hyperbola recedes from the center, its branches approach these asymptotes. The central rectangle of the hyperbola is centered at the origin with sides that pass through each vertex and co-vertex; it is a useful tool for graphing the hyperbola and its asymptotes. To sketch the asymptotes of the hyperbola, simply sketch and extend the diagonals of the central rectangle. See .
In this section, we will limit our discussion to hyperbolas that are positioned vertically or horizontally in the coordinate plane; the axes will either lie on or be parallel to the x- and y-axes. We will consider two cases: those that are centered at the origin, and those that are centered at a point other than the origin.

### Deriving the Equation of a Hyperbola Centered at the Origin

Let
  and
  be the foci of a hyperbola centered at the origin. The hyperbola is the set of all points
  such that the difference of the distances from
  to the foci is constant. See .
If
  is a vertex of the hyperbola, the distance from
  to
  is
  The distance from
  to
  is
  The difference of the distances from the foci to the vertex is
If
  is a point on the hyperbola, we can define the following variables:
By definition of a hyperbola,
  is constant for any point
  on the hyperbola. We know that the difference of these distances is
  for the vertex
  It follows that
  for any point on the hyperbola. As with the derivation of the equation of an ellipse, we will begin by applying the distance formula. The rest of the derivation is algebraic. Compare this derivation with the one from the previous section for ellipses.
This equation defines a hyperbola centered at the origin with vertices
  and co-vertices

### Writing Equations of Hyperbolas in Standard Form

Just as with ellipses, writing the equation for a hyperbola in standard form allows us to calculate the key features: its center, vertices, co-vertices, foci, asymptotes, and the lengths and positions of the transverse and conjugate axes. Conversely, an equation for a hyperbola can be found given its key features. We begin by finding standard equations for hyperbolas centered at the origin. Then we will turn our attention to finding standard equations for hyperbolas centered at some point other than the origin.

### Hyperbolas Centered at the Origin

Reviewing the standard forms given for hyperbolas centered at
  we see that the vertices, co-vertices, and foci are related by the equation
  Note that this equation can also be rewritten as
  This relationship is used to write the equation for a hyperbola when given the coordinates of its foci and vertices.

### Hyperbolas Not Centered at the Origin

Like the graphs for other equations, the graph of a hyperbola can be translated. If a hyperbola is translated
  units horizontally and
  units vertically, the center of the hyperbola will be
  This translation results in the standard form of the equation we saw previously, with
  replaced by
  and
  replaced by

Like hyperbolas centered at the origin, hyperbolas centered at a point
  have vertices, co-vertices, and foci that are related by the equation
  We can use this relationship along with the midpoint and distance formulas to find the standard equation of a hyperbola when the vertices and foci are given.

###  Graphing Hyperbolas Centered at the Origin

When we have an equation in standard form for a hyperbola centered at the origin, we can interpret its parts to identify the key features of its graph: the center, vertices, co-vertices, asymptotes, foci, and lengths and positions of the transverse and conjugate axes. To graph hyperbolas centered at the origin, we use the standard form
  for horizontal hyperbolas and the standard form
  for vertical hyperbolas.

###  Graphing Hyperbolas Not Centered at the Origin

Graphing hyperbolas centered at a point
  other than the origin is similar to graphing ellipses centered at a point other than the origin. We use the standard forms
  for horizontal hyperbolas, and
  for vertical hyperbolas. From these standard form equations we can easily calculate and plot key features of the graph: the coordinates of its center, vertices, co-vertices, and foci; the equations of its asymptotes; and the positions of the transverse and conjugate axes.

### Solving Applied Problems Involving Hyperbolas

As we discussed at the beginning of this section, hyperbolas have real-world applications in many fields, such as astronomy, physics, engineering, and architecture. The design efficiency of hyperbolic cooling towers is particularly interesting. Cooling towers are used to transfer waste heat to the atmosphere and are often touted for their ability to generate power efficiently. Because of their hyperbolic form, these structures are able to withstand extreme winds while requiring less material than any other forms of their size and strength. See . For example, a 500-foot tower can be made of a reinforced concrete shell only 6 or 8 inches wide!
The first hyperbolic towers were designed in 1914 and were 35 meters high. Today, the tallest cooling towers are in France, standing a remarkable 170 meters tall. In  we will use the design layout of a cooling tower to find a hyperbolic equation that models its sides.

### Key Equations

### Key Concepts

1. A hyperbola is the set of all points
  in a plane such that the difference of the distances between
  and the foci is a positive constant.
2. The standard form of a hyperbola can be used to locate its vertices and foci. See .
3. When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form. See  and .
4. When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola. See  and .
5. Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine whether the following equations represent hyperbolas. If so, write in standard form.
For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.
For the following exercises, find the equations of the asymptotes for each hyperbola.

### Graphical

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.
For the following exercises, given information about the graph of the hyperbola, find its equation.
For the following exercises, given the graph of the hyperbola, find its equation.

###  Extensions

For the following exercises, express the equation for the hyperbola as two functions, with
  as a function of
  Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes.

###  Real-World Applications

For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph.
For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information.


# Analytic Geometry

## The Parabola

### Learning Objectives

1. Graph vertical parabolas. (IA 11.2.1)
2. Graph horizontal parabolas. (IA 11.2.2)

### Objective 1: Graph vertical parabolas. (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.

Previously, we learned to graph vertical parabolas from the general form or the standard form using properties. Those methods will also work here.

### Practice Makes Perfect

Graph vertical parabolas.

### Objective 2: Graph horizontal parabolas. (IA 11.2.2)

Our work so far has only dealt with parabolas that open up or down. We are now going to look at horizontal parabolas. These parabolas open either to the left or to the right. If we interchange the x and y in our previous equations for parabolas, we get the equations for the parabolas that open to the left or to the right.

### Practice Makes Perfect

Katherine Johnson is the pioneering NASA mathematician who was integral to the successful and safe flight and return of many human missions as well as satellites. Prior to the work featured in the movie Hidden Figures, she had already made major contributions to the space program. She provided trajectory analysis for the Mercury mission, in which Alan Shepard became the first American to reach space, and she and engineer Ted Sopinski authored a monumental paper regarding placing an object in a precise orbital position and having it return safely to Earth. Many of the orbits she determined were made up of parabolas, and her ability to combine different types of math enabled an unprecedented level of precision.  Johnson said, "You tell me when you want it and where you want it to land, and I'll do it backwards and tell you when to take off."
Johnson's work on parabolic orbits and other complex mathematics resulted in successful orbits, Moon landings, and the development of the Space Shuttle program. Applications of parabolas are also critical to other areas of science. Parabolic mirrors (or reflectors) are able to capture energy and focus it to a single point. The advantages of this property are evidenced by the vast list of parabolic objects we use every day: satellite dishes, suspension bridges, telescopes, microphones, spotlights, and car headlights, to name a few. Parabolic reflectors are also used in alternative energy devices, such as solar cookers and water heaters, because they are inexpensive to manufacture and need little maintenance. In this section we will explore the parabola and its uses, including low-cost, energy-efficient solar designs.

### Graphing Parabolas with Vertices at the Origin

In The Ellipse, we saw that an ellipse is formed when a plane cuts through a right circular cone. If the plane is parallel to the edge of the cone, an unbounded curve is formed. This curve is a parabola. See .
Like the ellipse and hyperbola, the parabola can also be defined by a set of points in the coordinate plane. A parabola is the set of all points
       in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix.
In Quadratic Functions, we learned about a parabola’s vertex and axis of symmetry. Now we extend the discussion to include other key features of the parabola. See . Notice that the axis of symmetry passes through the focus and vertex and is perpendicular to the directrix. The vertex is the midpoint between the directrix and the focus.
The line segment that passes through the focus and is parallel to the directrix is called the latus rectum. The endpoints of the latus rectum lie on the curve. By definition, the distance
  from the focus to any point
  on the parabola is equal to the distance from
     to the directrix.
To work with parabolas in the coordinate plane, we consider two cases: those with a vertex at the origin and those with a vertex at a point other than the origin. We begin with the former.
Let
     be a point on the parabola with vertex
     focus
     and directrix

as shown in . The distance
  from point
     to point

on the directrix is the difference of the y-values:
     The distance from the focus
  to the point
     is also equal to
     and can be expressed using the distance formula.
Set the two expressions for
     equal to each other and solve for
     to derive the equation of the parabola. We do this because the distance from
     to
     equals the distance from
     to

We then square both sides of the equation, expand the squared terms, and simplify by combining like terms.
The equations of parabolas with vertex
     are
     when the x-axis is the axis of symmetry and
     when the y-axis is the axis of symmetry. These standard forms are given below, along with their general graphs and key features.
The key features of a parabola are its vertex, axis of symmetry, focus, directrix, and latus rectum. See . When given a standard equation for a parabola centered at the origin, we can easily identify the key features to graph the parabola.
A line is said to be tangent to a curve if it intersects the curve at exactly one point. If we sketch lines tangent to the parabola at the endpoints of the latus rectum, these lines intersect on the axis of symmetry, as shown in .

###  Writing Equations of Parabolas in Standard Form

In the previous examples, we used the standard form equation of a parabola to calculate the locations of its key features. We can also use the calculations in reverse to write an equation for a parabola when given its key features.

### Graphing Parabolas with Vertices Not at the Origin

Like other graphs we’ve worked with, the graph of a parabola can be translated. If a parabola is translated
       units horizontally and
       units vertically, the vertex will be
       This translation results in the standard form of the equation we saw previously with
       replaced by
       and
       replaced by

To graph parabolas with a vertex
       other than the origin, we use the standard form
       for parabolas that have an axis of symmetry parallel to the x-axis, and
       for parabolas that have an axis of symmetry parallel to the y-axis. These standard forms are given below, along with their general graphs and key features.

### Solving Applied Problems Involving Parabolas

As we mentioned at the beginning of the section, parabolas are used to design many objects we use every day, such as telescopes, suspension bridges, microphones, and radar equipment. Parabolic mirrors, such as the one used to light the Olympic torch, have a very unique reflecting property. When rays of light parallel to the parabola’s axis of symmetry are directed toward any surface of the mirror, the light is reflected directly to the focus. See . This is why the Olympic torch is ignited when it is held at the focus of the parabolic mirror.
Parabolic mirrors have the ability to focus the sun’s energy to a single point, raising the temperature hundreds of degrees in a matter of seconds. Thus, parabolic mirrors are featured in many low-cost, energy efficient solar products, such as solar cookers, solar heaters, and even travel-sized fire starters.

### Key Equations

### Key Concepts

1. A parabola is the set of all points
         in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix.
2. The standard form of a parabola with vertex
         and the x-axis as its axis of symmetry can be used to graph the parabola. If
         the parabola opens right. If
         the parabola opens left. See .
3. The standard form of a parabola with vertex
         and the y-axis as its axis of symmetry can be used to graph the parabola. If
         the parabola opens up. If
         the parabola opens down. See .
4. When given the focus and directrix of a parabola, we can write its equation in standard form. See .
5. The standard form of a parabola with vertex
         and axis of symmetry parallel to the x-axis can be used to graph the parabola. If
         the parabola opens right. If
         the parabola opens left. See .
6. The standard form of a parabola with vertex
         and axis of symmetry parallel to the y-axis can be used to graph the parabola. If
         the parabola opens up. If
         the parabola opens down. See .
7. Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides. See .

### Section Exercises

### Verbal

###  Algebraic

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form.
For the following exercises, rewrite the given equation in standard form, and then determine the vertex
         focus
         and directrix
  of the parabola.

### Graphical

For the following exercises, graph the parabola, labeling the focus and the directrix.
For the following exercises, find the equation of the parabola given information about its graph.
For the following exercises, determine the equation for the parabola from its graph.

###  Extensions

For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation.

###  Real-World Applications


# Analytic Geometry

## Rotation of Axes

### Learning Objectives

1. Using rotation of axes formulas.
2. Identify conic sections by their equations. (IA 11.4.3)

### Objective 1: Using rotation of axes formulas.

If a point

on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle

from the positive
x
-axis, then the coordinates of the point with respect to the new axes are

The following rotations of axes formulas define the relationship between (x,y) and (x’,y’):

### Practice Makes Perfect

Using rotation of axes formulas:

### Objective 2: Identify conic sections by their equations. (IA 11.4.3)

We can identify a conic from its equations by looking at the signs and coefficients of the variables that are squared.

### Practice Makes Perfect

Identify conic sections by their equations.

As we have seen, conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions, which we also call a cone. The way in which we slice the cone will determine the type of conic section formed at the intersection. A circle is formed by slicing a cone with a plane perpendicular to the axis of symmetry of the cone. An ellipse is formed by slicing a single cone with a slanted plane not perpendicular to the axis of symmetry. A parabola is formed by slicing the plane through the top or bottom of the double-cone, whereas a hyperbola is formed when the plane slices both the top and bottom of the cone. See .
Ellipses, circles, hyperbolas, and parabolas are sometimes called the nondegenerate conic sections, in contrast to the degenerate conic sections, which are shown in . A degenerate conic results when a plane intersects the double cone and passes through the apex. Depending on the angle of the plane, three types of degenerate conic sections are possible: a point, a line, or two intersecting lines.

### Identifying Nondegenerate Conics in General Form

In previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections. In this section, we will shift our focus to the general form equation, which can be used for any conic. The general form is set equal to zero, and the terms and coefficients are given in a particular order, as shown below.
where
    and
    are not all zero. We can use the values of the coefficients to identify which type conic is represented by a given equation.
You may notice that the general form equation has an
    term that we have not seen in any of the standard form equations. As we will discuss later, the
    term rotates the conic whenever
  is not equal to zero.

### Finding a New Representation of the Given Equation after Rotating through a Given Angle

Until now, we have looked at equations of conic sections without an
     term, which aligns the graphs with the x- and y-axes. When we add an
     term, we are rotating the conic about the origin. If the x-  and y-axes are rotated through an angle, say
     then every point on the plane may be thought of as having two representations:
     on the Cartesian plane with the original x-axis and y-axis, and
  on the new plane defined by the new, rotated axes, called the x'-axis and y'-axis. See .
We will find the relationships between
     and
     on the Cartesian plane with
     and
     on the new rotated plane. See .
The original coordinate x- and y-axes have unit vectors
     and
     The rotated coordinate axes have unit vectors
     and
     The angle
     is known as the angle of rotation. See . We may write the new unit vectors in terms of the original ones.
Consider a vector in the new coordinate plane. It may be represented in terms of its coordinate axes.
Because
     we have representations of
     and
     in terms of the new coordinate system.

### Writing Equations of Rotated Conics in Standard Form

Now that we can find the standard form of a conic when we are given an angle of rotation, we will learn how to transform the equation of a conic given in the form
    into standard form by rotating the axes. To do so, we will rewrite the general form as an equation in the
    and
    coordinate system without the
    term, by rotating the axes by a measure of
    that satisfies
We have learned already that any conic may be represented by the second degree equation
where
    and
    are not all zero. However, if
    then we have an
    term that prevents us from rewriting the equation in standard form. To eliminate it, we can rotate the axes by an acute angle
    where

###  Identifying Conics without Rotating Axes

Now we have come full circle. How do we identify the type of conic described by an equation? What happens when the axes are rotated? Recall, the general form of a conic is
If we apply the rotation formulas to this equation we get the form
It may be shown that
    The expression does not vary after rotation, so we call the expression invariant. The discriminant,
    is invariant and remains unchanged after rotation. Because the discriminant remains unchanged, observing the discriminant enables us to identify the conic section.

### Key Equations

### Key Concepts

1. Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail. They include an ellipse, a circle, a hyperbola, and a parabola.
2. A nondegenerate conic section has the general form
  	   where
  	   and
  	   are not all zero. The values of
  	   and
  	   determine the type of conic. See .
3. Equations of conic sections with an
  	   term have been rotated about the origin. See .
4. The general form can be transformed into an equation in the
  	   and
  	   coordinate system without the
  	   term. See  and .
5. An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant, observing it enables us to identify the conic section. See .

### Section Exercises

### Verbal

### Algebraic

For the following exercises, determine which conic section is represented based on the given equation.
For the following exercises, find a new representation of the given equation after rotating through the given angle.
For the following exercises, determine the angle
     that will eliminate the
     term and write the corresponding equation without the
     term.

### Graphical

For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation.
For the following exercises, graph the equation relative to the
     system in which the equation has no
     term.
For the following exercises, determine the angle of rotation in order to eliminate the
     term. Then graph the new set of axes.
For the following exercises, determine the value of
     based on the given equation.