Lesson 12
Solutions to Linear Equations
12.1: Estimate Area (5 minutes)
Warmup
This warmup prompts students to reason about the area of triangles and quadrilaterals. If there is time after sharing the estimates and reasoning, ask the students for the information they would need to get an exact answer to this question and how they would use that information.
Launch
Ensure students know what is meant by the shaded region. Display the figures. Give students 1 minute of quiet think time and ask them to give a signal when they have an answer and reasoning. Encourage students to have a reasoning to defend their choice beyond, “It looks like more.” Follow with a wholeclass discussion.
Student Facing
Which figure has the largest shaded region?
Student Response
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Activity Synthesis
Invite students to share how they visualized the shaded region of each figure. Record and display their explanations for all to see. Solicit from the class alternative ways of quantifying the shaded portion and alternative ways of naming the size of the shaded portion (to elicit representations as fractions, decimals, or percentages). To involve more students in the conversation, consider asking:
 “Who can restate ___’s reasoning in a different way?”
 “Did anyone solve the shaded portion the same way but would explain it differently?”
 “Did anyone solve the shaded portion in a different way?”
 “Does anyone want to add on to _____’s strategy?”
 “Do you agree or disagree? Why?”
12.2: Apples and Oranges (15 minutes)
Activity
In previous lessons in this unit, students have analyzed multiple situations that lead to equations of the form \(y = mx + b\), including positive, negative, and 0 slope. In the previous lesson, they saw that vertical lines cannot be described by equations of this form and saw a geometric situation that could be represented by an equation of the form \(Ax + By = C\). This form of a linear equation is examined in greater detail here as students consider combinations of numbers that keep a total cost constant and write an equation that will be used again in the next activity. In this problem, the solutions are limited to nonnegative integers and the set of all solutions is finite. In the next activity, the solutions comprise all real numbers and the set of solutions is infinite.
There are many equations students could use to describe the cost of apples in dollars, \(a\), and the cost of oranges in dollars, \(r\). Identify students who write \(a + 2r = 10\), and ask them to share during the discussion. There is no reason to solve for one variable in terms of the other, in this case, because a graph has not been requested and neither variable is a “natural” candidate for dependent versus independent. In this activity, students need to explain their reasoning (MP3).
Launch
Arrange students in groups of 2. Give students 3–5 minutes of quiet think time to answer the first question and think about the others. Have partners compare solutions and discuss the remaining questions. Follow with a wholeclass discussion.
Supports accessibility for: Language; Socialemotional skills
Design Principle(s): Support sensemaking; Maximize metaawareness
Student Facing
At the corner produce market, apples cost $1 each and oranges cost $2 each.

Find the cost of:
 6 apples and 3 oranges
 4 apples and 4 oranges
 5 apples and 4 oranges
 8 apples and 2 oranges

Noah has $10 to spend at the produce market. Can he buy 7 apples and 2 oranges? Explain or show your reasoning.

What combinations of apples and oranges can Noah buy if he spends all of his $10?

Use two variables to write an equation that represents $10combinations of apples and oranges. Be sure to say what each variable means.

What are 3 combinations of apples and oranges that make your equation true? What are three combinations of apples and oranges that make it false?
Student Response
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Student Facing
Are you ready for more?
 Graph the equation you wrote relating the number of apples and the number of oranges.
 What is the slope of the graph? What is the meaning of the slope in terms of the context?
 Suppose Noah has $20 to spend. Graph the equation describing this situation. What do you notice about the relationship between this graph and the earlier one?
Student Response
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Anticipated Misconceptions
Students may write an equation like \(a + r = 10\) where \(a\) is the cost of the apples and \(r\) is the cost of the oranges. While this is technically correct, it leaves extra work figuring out whether or not a solution to this corresponds to some actual number of apples and oranges. Ask these students what they are trying to figure out (numbers of apples and numbers of oranges that cost $10). Then suggest that they use their variables to represent the number of apples and the number of oranges rather than their cost.
Activity Synthesis
The equation \(a + 2r = 10\) (or some equivalent) is an example of a naturally arising equation of the form \(Ax + By = C\), introduced in the previous lesson. In a graphical representation of this situation, either variable (oranges or apples) could be plotted on either axis. If students mention slope, make sure to ask them how they are plotting the variables. This will be taken up in greater detail in the next activity where the graphical representation will be central.
Invite students from different groups to share what they discovered with their partners by asking:

“How many combinations that cost $10 did you find in total? How do you know you have found them all?” (6 combinations, these are all the possibilities of whole numbers that would make the sum 10.)

“Did you notice any patterns that helped you find the combination?” (There are many patterns; the one we would like students to notice is that buying 1 less orange means you can buy 2 more apples.)

“What would the graph of the equation you wrote look like?” (Some may realize that the graph would contain the six points that represent the six combinations for $10. Help them to see that these points would all lie on a line, given by the equation \(a + 2r = 10\). The noninteger points on the line would represent a fractional number of pieces of each fruit.)
 “According to the equation you wrote, if you bought \(\frac{1}{2}\) orange and 9 apples you would spend $10. Do you think this situation is realistic when buying fruit in a store?” (Probably not, you would only buy whole pieces of fruit.)
12.3: Solutions and Everything Else (15 minutes)
Activity
Students write an equation representing a stated relationship between two quantities, and use the equation to find pairs of numbers that make it true and pairs of numbers for which it is not true. By graphing both sets of points, students see that the graph of a linear equation is the set of its solutions, that is, the points whose coordinates make the equation true.
Students should be encouraged to use rational number coordinates and points in all four quadrants and on the two axes in their graphs, and to see that solutions exist all along the line, between and beyond the pairs they explicitly choose, making for an infinite number of solutions.
Some students might notice that the points where \(x=0\) and \(y=0\) are easy to quickly find and that connecting these two points yields the line containing all the other solutions. They may then try to read solutions from the graph instead of using the equation to more accurately calculate the coordinates of solutions. While this is a valuable observation in terms of quickly drawing the graph of the linear equation, students should be guided to realize that the equation enables a level of precision that reading from the graph often lacks. Precision can be improved through the use of technology in the form of graphing calculators and applications.
As students find points, both satisfying the relationship from the first question and not satisfying the relationship, monitor for these choices and invite these students to share during the discussion.
 Points in the first quadrant
 Points on the axes (there are only two of these!)
 Points in the first and/or fourth quadrants
 Points with noninteger coordinate values
Launch
Students should have access to geometry toolkits, especially rulers and graph paper.
Display for all to see the following statement: “You have two numbers. If you double the first number and add it to the second number, the sum is 10.”
Poll the class for “predict how many pairs of numbers make this statement true?” Display the answers for all to see.
Instruct students to pause their work after creating their graph and then to check their graph with you. Identify students who struggle with writing the equation or with finding and graphing solutions. Check that the graph and the equation match since some students might write the equation as \(x + 2y = 10\) instead of \(2x + y = 10\). Also check that students are not restricting \(x\) and \(y\) values to nonnegative integers.
Supports accessibility for: Conceptual processing
Student Facing
You have two numbers. If you double the first number and add it to the second number, the sum is 10.

Let \(x\) represent the first number and let \(y\) represent the second number. Write an equation showing the relationship between \(x\), \(y\), and 10.

Draw and label a set of \(x\) and \(y\)axes. Plot at least five points on this coordinate plane that make the statement and your equation true. What do you notice about the points you have plotted?

List ten points that do not make the statement true. Using a different color, plot each point in the same coordinate plane. What do you notice about these points compared to your first set of points?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
Students might write \(x + 2y = 10\). Ask these students what \(x\) and \(y\) represent and to pick a couple of values of \(x\) and \(y\) so that the double of \(x\) and \(y\) sum to 10. Ask them to see if these are solutions to their equation. They most likely will not be. Suggest that these students reconsider their equation.
Activity Synthesis
Select students to share their examples of points in the coordinate plane that make their equation true, sequenced in the following order:
 Points in the first quadrant
 Points on the axes (there are only two of these!)
 Points in the first and/or fourth quadrants
 Points with noninteger coordinate values
If it does not come up in student work, ask them if the equation could be true if \(x = \text3\) (yes, \(y = 16\)) or if \(y = \text4\) (yes, \(x = 7\)). What if \(x = 3\frac{1}{2}\)? (Yes, \(y = 3\).) Through the discussion questions, bring out that there are an infinite number of solutions to this equation and that, taken collectively and plotted in the plane, they make up the line of all solutions to the equation \(2x+y = 10\).
Connect student solutions to the graphical representation of the scenario by asking:
 “For an equation with two variables, a pair of numbers that makes the equation true is called a solution of the equation. Where in your graph do you see solutions of the equation that you wrote? Are all possible solutions on your graph?” Bring out here that the set of all solutions to the equation is a line with \(y\)intercept 10 and slope 2. So an alternative way to write its equation, other than \(2x + y = 10\), is \(y = \text2x + 10\).

“Where in your graph do you see pairs that are not solutions of the equation you wrote?” (All points not on the line.)

“Based on your observations, what is the relationship between the solutions of an equation and its graph?” (For this equation, the set of solutions is a line.)

“What does the graph tell you about the number of solutions of your equation?” (There are an infinite number.)

“Does the line contain all possible solutions to your equation? How do you know?” (Yes, for each value of \(x\), there is exactly one value of \(y\) giving a solution.)
In summary, articulate the idea that the solutions are precisely the points on the line, and no more.
Ask students to compare solutions to the apple and orange activity with solutions in this activity. “Would a graph of all solutions to the apples and oranges problem look the same as the graph in this activity?” It would not because you can only purchase nonnegative whole number values of apples and oranges. Make sure students understand that in this activity it makes sense to draw a line connecting all points because the coordinates of all the points on the line are solutions to the equation.
A solution to an equation with two variables is an ordered pair of values that makes an equation true, and the graph of the equation is the set of all solution pairs \((x,y)\) plotted as points in coordinate plane: for \(2x + y = 10\), this set of solutions is a line with slope 2 and \(y\)intercept 10.
Design Principle(s): Optimize output ; Cultivate conversation
Lesson Synthesis
Lesson Synthesis
Students explored several big ideas in this lesson:
 A solution to a linear equation is a pair of values that makes the equation true.
 Solutions can be found by substituting a value for one of the variables and solving the equation for the other.
 The set of all the solutions to a linear equation can be shown visually in the coordinate plane and is called the graph of the equation.
 The graph of a linear equation is a line.
 Any points in the coordinate plane that do not lie on the line that is on the graph of the linear equation are not solutions to the equation.
 The number of solutions might be limited in a realworld situation even though the equation has an infinite number of solutions.
Invite students to describe how they found solutions to linear equations and to explain how they knew they had found a solution.
Ask what difference there may be between reading solutions from a graph and calculating them using the equation. Listen for students to identify a possible lack of precision when reading from a graph. If you are using graphing software this will be less of an issue.
Finally, ask if points that are not on the line can be solutions to the equation represented by the line. Listen for students to understand that the line represents all pairs that make the equation true so a point not on the line cannot be a solution.
12.4: Cooldown  Identify the Points (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
Think of all the rectangles whose perimeters are 8 units. If \(x\) represents the width and \(y\) represents the length, then \(\displaystyle 2x+2y=8\) expresses the relationship between the width and length for all such rectangles.
For example, the width and length could be 1 and 3, since \(2 \boldcdot 1 + 2 \boldcdot 3 = 8\) or the width and length could be 2.75 and 1.25, since \(2 \boldcdot (2.75) + 2 \boldcdot (1.25) = 8\).
We could find many other possible pairs of width and length, \((x,y)\), that make the equation true—that is, pairs \((x,y)\) that when substituted into the equation make the left side and the right side equal.
A solution to an equation with two variables is any pair of values \((x,y)\) that make the equation true.
We can think of the pairs of numbers that are solutions of an equation as points on the coordinate plane. Here is a line created by all the points \((x,y)\) that are solutions to \(2x+2y=8\). Every point on the line represents a rectangle whose perimeter is 8 units. All points not on the line are not solutions to \(2x+2y=8\).