options: Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.
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: 4