Lesson 11
More Solutions to Linear Equations
11.1: Coordinate Pairs (5 minutes)
Warmup
The purpose of this warmup is for students to practice solving an equation for an unknown value while thinking about a coordinate pair, \((x,y)\), that makes the equation true. While the steps to solve the equation are the same regardless of which value of \(x\) students choose, there are strategic choices that make solving the resulting equation simpler. This should be highlighted in the discussion.
Launch
Encourage students to not pick 0 for \(x\) each time.
Student Facing
For each equation choose a value for \(x\) and then solve to find the corresponding \(y\) value that makes that equation true.
 \(6x=7y\)
 \(5x+3y=9\)
 \(y+5\frac13 x=7\)
Student Response
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Activity Synthesis
Collect the pairs of \(x\)’s and \(y\)’s students calculated and graph them on a set of axes. For each equation, they form a different line. Have students share how they picked their \(x\) values. For example:
 For the first problem, choosing \(x\) to be a multiple of 7 makes \(y\) an integer.
 For the last problem, picking \(x\) to be a multiple of 3 makes \(y\) an integer.
11.2: True or False: Solutions in the Coordinate Plane (15 minutes)
Activity
In the previous lesson, students studied the set of solutions to a linear equation, the set of all values of \(x\) and \(y\) that make the linear equation true. They identified that this was a line in the coordinate plane. In this activity, they are given graphs of lines and then are asked whether or not different \(x\)\(y\) coordinate pairs are solutions to equations that define the lines. This helps students solidify their understanding of the relationship between a linear equation and its graph in the coordinate plane.
Consider asking students to work on the first 5 problems only if time is an issue. Since this activity largely reinforces the material of the previous lesson, it is not essential to do all 8 problems.
Launch
Arrange students in groups of 2. Students work through the eight statements individually and then compare and discuss their answers with their partner. Tell students that if they disagree, they should work to come to an agreement.
Supports accessibility for: Organization; Attention
Student Facing
Here are graphs representing three linear relationships. These relationships could also be represented with equations.
For each statement below, decide if it is true or false. Explain your reasoning.

\((4,0)\) is a solution of the equation for line \(m\).

The coordinates of the point \(G\) make both the equation for line \(m\) and the equation for line \(n\) true.

\(x = 0\) is a solution of the equation for line \(n\).

\((2,0)\) makes both the equation for line \(m\) and the equation for line \(n\) true.

There is no solution for the equation for line \(\ell\) that has \(y = 0\).

The coordinates of point \(H\) are solutions to the equation for line \(\ell\).

There are exactly two solutions of the equation for line \(\ell\).

There is a point whose coordinates make the equations of all three lines true.
After you finish discussing the eight statements, find another group and check your answers against theirs. Discuss any disagreements.
Student Response
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Anticipated Misconceptions
Some students may try to find equations for the lines. There is not enough information to accurately find these equations, and it is not necessary since the questions only require understanding that a coordinate pair lies on a line when it gives a solution to the corresponding linear equation. Ask these students if they can answer the questions without finding equations for the lines.
Activity Synthesis
Display the correct answer to each question, and give students a few minutes to discuss any discrepancies with their partner. For the third question, some students might say yes because there is a solution to the equation for line \(n\), which has \(x = 0\), namely if \(y = 0\) as well. For the fifth question, make sure students understand that the line \(\ell\) meets the \(x\)axis even if that point is not shown on the graph.
Some key points to highlight, reinforcing conclusions from the previous lesson as well as this activity:
 A solution of an equation in two variables is an ordered pair of numbers.
 Solutions of an equation lie on the graph of the equation.
Design Principle(s): Cultivate conversation; Maximize metaawareness
11.3: I’ll Take an X, Please (15 minutes)
Activity
Students are given linear equations—some of which represent proportional relationships—in various forms, and are also given solutions to their partner’s equations in the form of coordinates of a point. The student with the equation decides which quantity they would like to know, \(x\) or \(y\), and requests this information from their partner. They then solve for the other quantity. The activity reinforces the concept that solutions to equations with two variables are a pair of numbers, and that knowing one can give you the other by using the value you know and solving the equation. Students also have a chance to think about the most efficient way to find solutions for equations in different forms.
You will need the I’ll Take An \(x\) Please blackline master for this activity.
Launch
Consider demonstrating the first step with a student. Write the equation \(y = 5x  11\) on one slip of paper and the point \((1, \text 6)\) on another slip of paper. Give the slip with the coordinate pair to the student. You can ask for the \(x\) or \(y\) coordinate of a point on the graph and then need to find the other coordinate. Ask the student helper for either \(x\) or \(y\). (In this case, asking for the \(x\) coordinate is wise because then you can just plug it into the equation to find the corresponding \(y\)coordinate for the point on the graph.) Display your equation for all to see and demonstrate substituting the value in and solving for the other variable. Alternatively, ask students how they might use the information given (one of the values for \(x\) or \(y\) to find the other given your equation.
Arrange students in groups of 2. One partner receives Cards A through F from the left side of the blackline master and the other receives Cards a through f from the right side. Students take turns asking for either \(x\) or \(y\) then solving their equation for the other, and giving their partner the information requested.
Students play three rounds, where each round consists of both partners having a turn to ask for a value and to solve their equation. Follow with a wholeclass discussion.
Supports accessibility for: Language; Memory .
Student Facing
One partner has 6 cards labeled A through F and one partner has 6 cards labeled a through f. In each pair of cards (for example, Cards A and a), there is an equation on one card and a coordinate pair, \((x,y)\), that makes the equation true on the other card.

The partner with the equation asks the partner with a solution for either the \(x\)value or the \(y\)value and explains why they chose the one they did.

The partner with the equation uses this value to find the other value, explaining each step as they go.

The partner with the coordinate pair then tells the partner with the equation if they are right or wrong. If they are wrong, both partners should look through the steps to find and correct any errors. If they are right, both partners move onto the next set of cards.

Keep playing until you have finished Cards A through F.
Student Response
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Student Facing
Are you ready for more?
Consider the equation \(ax + by = c\), where \(a, b,\) and \(c\) are positive numbers.
 Find the coordinates of the \(x\) and \(y\)intercepts of the graph of the equation.
 Find the slope of the graph.
Student Response
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Activity Synthesis
The discussion should focus on using the given information to efficiently find a solution for the equation. Consider asking students:
 “How did you decide whether you wanted the value of \(x\) or the value of \(y\)?” (One might be more efficient to solve for: for example, with card B asking for \(x\) makes sense while with card d the arithmetic to perform is similar whether asking for \(x\) or for \(y\).)
 “Which equations represent proportional relationships? How do you know? Which do not?” (C and F are proportional because they can be written as \(y=kx\), although this is hidden at first glance in C.)
 “Once you have identified one solution to your equation, what are some ways you could find others?” (Use the constant rate of change to add/subtract to the solution you know, solve the equation for \(x\) or \(y\), choose values for one variable and solve for the other, for proportional relationships you could find equivalent ratios.)
Point out that all of the equations in this activity are linear. They are given in many different forms, not just \(y = mx + b\) or \(Ax + By = C\).
Design Principle(s): Support sensemaking; Cultivate conversation (for explanation)
11.4: Making Signs (10 minutes)
Activity
In the previous activity, the system of equations was represented in words, a table, and a graph. In this activity, the system of equations is partially given in words, but key elements are only provided in the graph. Students have worked with lines that represent a context before. Now they must work with two lines at the same time to determine whether a point lies on one line, both lines, or neither line.
Launch
Arrange students in groups of 2. Give students 1 minute to read the problem and answer any questions they have about the context. Tell students to complete the table one row at a time with one person responding for Clare and the other responding for Andre. Give students 2–3 minutes to finish the table followed by wholeclass discussion.
Supports accessibility for: Organization; Attention
Design Principle(s): Cultivate conversation; Maximize metaawareness
Student Facing
Clare and Andre are making signs for all the lockers as part of the decorations for the upcoming spirit week. Yesterday, Andre made 15 signs and Clare made 5 signs. Today, they need to make more signs. Each person's progress today is shown in the coordinate plane.
Based on the lines, mark the statements as true or false for each person.
point  what it says  Clare  Andre 

\(A\)  At 40 minutes, I have 25 signs completed.  
\(B\)  At 75 minutes, I have 42 and a half signs completed.  
\(C\)  At 0 minutes, I have 15 signs completed.  
\(D\)  At 100 minutes, I have 60 signs completed. 
Student Response
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Student Facing
Are you ready for more?
 4 toothpicks make 1 square
 7 toothpicks make 2 squares
 10 toothpicks make 3 squares
Do you see a pattern? If so, how many toothpicks would you need to make 10 squares according to your pattern? Can you represent your pattern with an expression?
Student Response
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Activity Synthesis
Display the graphs from the task statement. The goal of this discussion is for students to realize that points that lie on one line can be interpreted as statements that are true for Clare, and points that lie on the other line as statements that are true for Andre. Ask students:
 “What is true for Clare and Andre after 20 minutes?” (Clare has 15 signs completed and Andre has 20 signs completed.)
 “What, then, do you know about the point \((20, 15)\) and the equation for Clare's graph?” (The point \((20, 15)\) is a solution to the equation for Clare's graph.)
 “What do you know about the point \((20, 20)\) and the equation for Andre's graph?” (The point \((20, 20)\) is a solution to the equation for Andre's graph.)
Invite groups to share their reasoning about points A–D. Conclude by pointing out to students that, in this context, there are many points true for Clare and many points true for Andre but only one point true for both of them. Future lessons will be about how to figure out that point.
Lesson Synthesis
Lesson Synthesis
In order to highlight student thinking about different strategies for finding a solution to a linear equation, ask students:
 “What are different ways to find a solution to the linear equation \(3y + x = 12\)?” (Substitute in a value for one variable and solve for the other; graph the equation and find points that lie on the line; rearrange the equation so that one variable is written in terms of the other variable.)
 “How do you know when you have found a solution to the equation \(3y + x = 12\)?” (The coordinates of the point will make the statement true.)
 “What are some easy values to substitute into the equation?” (In this case, a good strategic choice is \(x = 0\), which gives \(y = 4\), and \(y=0\), which gives \(x=12\). This says that the \(y\)intercept of the graph of the equation is \((0,4)\). Similarly, the \(x\)intercept of the equation’s graph is \((12,0)\).
 “How can you find the slope of the line?” (Graphing the line shows that the slope is negative, and we can verify this by rewriting the equation as \(y = \text\frac{1}{3}x + 4\).)
11.5: Cooldown  Intercepted (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
Let's think about the linear equation \(2x4y=12\). If we know \((0,\text3)\) is a solution to the equation, then we also know \((0,\text3)\) is a point on the graph of the equation. Since this point is on the \(y\)axis, we also know that it is the vertical intercept of the graph. But what about the coordinate of the horizontal intercept, when \(y=0\)? Well, we can use the equation to figure it out.
\(\begin{align}2x4y&=12 \\2x4(0)&=12 \\2x&=12\\x&=6 \end{align}\)
Since \(x=6\) when \(y=0\), we know the point \((6,0)\) is on the graph of the line. No matter the form a linear equation comes in, we can always find solutions to the equation by starting with one value and then solving for the other value.