options: Represent and solve equations and inequalities graphically.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Work with radicals and integer exponents.

Use the four operations with whole numbers to solve problems.

Multiply and divide within 100.

Use polynomial identities to solve problems.

Write expressions in equivalent forms to solve problems.

Understand and apply properties of operations and the relationship between addition and subtraction.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Work with addition and subtraction equations.

Gain familiarity with factors and multiples.

Represent and solve problems involving multiplication and division.

Rewrite rational expressions.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Add and subtract within 20.

Create equations that describe numbers or relationships

Write and interpret numerical expressions.

Work with equal groups of objects to gain foundations for multiplication.

Interpret the structure of expressions.

Understand the connections between proportional relationships, lines, and linear equations.

Analyze patterns and relationships.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Perform arithmetic operations on polynomials.

Use properties of operations to generate equivalent expressions.

Generate and analyze patterns.

Solve equations and inequalities in one variable.

Reason about and solve one-variable equations and inequalities.

Represent and analyze quantitative relationships between dependent and independent variables.

Understand properties of multiplication and the relationship between multiplication and division.

Solve systems of equations.

Understand the relationship between zeros and factors of polynomials.

Represent and solve problems involving addition and subtraction.

Understand solving equations as a process of reasoning and explain the reasoning.
problem_activity: Activity
In this activity, students solve a system of equations by graphing then include a third line through the solution point that is either vertical or horizontal. In the associated Algebra lesson, students will graph equations in a system and additional equations that arise from the elimination method to show that they all intersect at the solution.
To focus on the work of the activity, it is recommended that students use the digital version of the activity or have access to Desmos or another graphing tool to more quickly and accurately find the solutions to the system of equations through graphing. If students need practice graphing lines representing equations by hand or if digital access is not available, the print version provides a graph given one of the lines and asks students to graph the second line by hand.
Launch
Display the instructions for all to see. Ask students,
"What do you know about equations that are represented by vertical lines?" (They are of the form
\(x = a\)
for a number
\(a\)
.)
"What do you know about equations that are represented by horizontal lines?" (They are of the form
\(y = a\)
for a number
\(a\)
.)
Student Facing
For each system of equations:
Solve the system of equations by graphing. Write the solution as an ordered pair.
Write an equation that would represented by a vertical or horizontal line that also passes through the solution of the system of equations.
Graph your new equation along with the system.
\(\begin {cases} \begin {align}y = 3x+5\\ y=\text{-}x+1\end{align} \end {cases}\)
The line representing
\(y = 3x+5\)
is shown
[IMG]
\(\begin {cases} \begin {align}y = \frac{1}{3}x-2\\ y=x-6\end{align} \end {cases}\)
The line representing
\(y = \frac{1}{3}x-2\)
is shown
[IMG]
\(\begin {cases} \begin {align} 2x+3y=10\\ x+y=3\end{align} \end {cases}\)
The line representing
\(2x+3y=10\)
is shown
[IMG]
Student Response
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Activity Synthesis
The purpose of the discussion is for students to recognize that, for a system of equations represented by graphs that intersect at
\((a,b)\)
, the lines
\(x = a\)
and
\(y = b\)
also go through the solution. Select students to share their solutions including students who found vertical as well as horizontal lines through the intersection point.
Some questions for discussion:
"How could you use your answers to these questions to write another system of equations that have the same solution as the original system of equations?" (The system
\(\begin {cases} \begin {align} x = \text{-}1\\ y=2\end{align} \end {cases}\)
has the same solution as the original equation
\(\begin {cases} \begin {align}y = 3x+5\\ y=\text{-}x+1\end{align} \end {cases}\)
since all 4 lines go through the point
\((\text{-}1,2)\)
.)
"Two lines intersect at the point
\((3,5)\)
. How could you know whether the equation
\(3x - 2y = \text{-}1\)
is represented by a line that goes through that same point?" (Substitute the value 3 for
\(x\)
and 5 for
\(y\)
. If the equation is still true, then the associated line will go through the same point. If the equation is false, the associated line will not go through the same point.)


correct_option_index: 30