Lesson 17

In grade 8, students learned that some systems of linear equations have many solutions. This warm-up reminds students about this fact, while also prompting them to use what they learned in this unit to better understand what it means for a system to have infinitely many solutions.

The first equation shows two variables adding up to 3, so students choose a pair of values whose sum is 3. They notice that all pairs chosen are solutions to the system. Next, they try to find a strategy that can show that there are countless other pairs that also satisfy the constraints in the system. Monitor for these likely strategies:

• Solving by graphing: The graphs of the two equations are the same line, so all the points on the line are solutions to the system.
• Solving by substitution: The first equation can be rearranged to \(y = 3 - x\). Substituting \(3-x\) for \(y\) in the second equation gives \(4x = 12 - 4(3-x)\) or \(4x = 12-12+4x\) or \(4x = 4x\). This equation is true no matter what \(x\) is.
• Solving by elimination: If we multiply the first equation by 4, rearrange the second equation to \(4x+4y=12\), and then subtract the second equation from the first, the result is \(4y=4y\). Subtracting \(4y\) from each side gives \(0=0\), which is true regardless of what \(x\) or \(y\) is.
• Reasoning about equivalent equations: If we rearrange the second equation so that the variables are on the same side \(4x+4y=12\), we can see that this equation is a multiple of the first and are equivalent. This means they have the exact same solution set, which contains infinite possible pairs of \(x\) and \(y\).

Identify students using different strategies and ask them to share their thinking with the class later.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Arrange students in groups of 3–4. Give students 1–2 minutes of quiet time to complete the first two questions. Remind students that the numbers don't have be integers. Then, give students a minute to share with their group what their two values are and whether the pair is a solution to the second equation.

Pause for a brief whole-class discussion. Consider recording quickly all the pairs that students have verified to be solutions to the second equation and displaying them for all to see. Ask: "Which two numbers are the solution to the system?" (All of them)

Ask students to proceed to the last question. Students may conclude that seeing all the pairs that satisfy both equations is enough to show that the system has many solutions. If so, ask how many solutions they think there are, and ask: "Can you show that any two numbers that add up to 3 is also a solution to the system, without having to do calculations for each pair?" 

Ask the class how many solutions they think the system has. Select previously identified students to explain how they know that the system has infinitely many solutions, or that any pair of values that add up to 3 and make the first equation true also make the second equation true.

Students are likely to think of using graphs. It is not essential to elicit all strategies shown in the Activity Narrative, but if no one thinks of an algebraic explanation, be sure to bring one up. The last strategy mentioned in the Narrative (reasoning about equivalent equations) is likely to be intuitive to students.

Make sure students see that the equations are equivalent, so all pairs of \(x\) and \(y\) values that make one equation true also make the other equation true, giving an infinite number of solutions.

In the warm-up, students saw that some systems have infinitely many solutions. In this activity, they encounter a situation that can be represented with a system of equations but the system has no solutions. Students write equations to represent the two constraints in the situation and then solve the system algebraically and graphically.

As students work, notice the different ways they reach the conclusion that the systems have no solutions. Identify students with varying strategies and ask them to share later.

Keep students in groups of 3–4 and provide access to graphing technology.

Invite previously identified students to share their response to the second question. Record or display their reasoning for all to see. After each student shares, ask if anyone else reasoned the same way.

Next, select other students to share their observations about the graphs. Ask students: 

• “How can we tell from the graphs that there are no solutions?” (The lines are parallel.)
• “How can we tell for sure that the lines are parallel and never intersect?” (The slope of both graphs is -2 but they have different intercepts.)
• “Why do parallel lines mean no solutions?” (A solution is a pair of values that satisfy both equations and are on both graphs. There are no points that are on both lines simultaneously.)
• “What does ‘no solutions’ mean in this situation, in terms of price of pool passes and gym memberships?” (The prices for a pool pass and for a gym membership are different for the two purchases.)

Here are some ways to think about the situation: 

• The family purchased twice the number of of pool passes and gym memberships as the individual did, but they did not pay twice as much, so the prices of passes and memberships must have been different for the two purchases.
• The person who bought half as many passes and memberships did not pay half as much, which meant that different prices applied to the two transactions.
• The special rates for a family of 4 did not apply to individuals, hence the different prices.

In earlier activities, students gained some insights on the structure of equations in systems that have infinitely many solutions and those that have no solutions. In this activity, they apply those insights to sort systems of equations based on the number of solutions (one solution, many solutions, or no solutions).

Students could solve each system algebraically or graphically and sort afterwards, but given the number of systems to be solved, they will likely find this process to be time consuming. A more productive way would be to look for and make use of the structure in the equations in the systems (MP7), for example, by looking out for equivalent equations, equations with the same slope but different vertical intercepts, variable expressions with the same or opposite coefficients, and so on.

To effectively make use of the structure of the systems, students need to attend closely to all parts of each equation—the signs, variables, coefficients and constants—and to rearrange equations with care (MP6).

As students discuss their thinking in groups, make note of the different ways they use structure to complete the task. Encourage students who are solving individual systems to analyze the features of the equations and see if they could reason about the solutions or gain information about the graphs that way.

In this activity, students are analyzing the structure of equations in the systems, so technology is not an appropriate tool.

Here is an image of the cards for reference and planning:

Arrange students in groups of 2. Give one set of pre-cut slips or cards from the blackline master to each group.

Give students 7–8 minutes to sort the cards into groups. Emphasize to students that they should be prepared to explain how they place each system. Follow with a whole-class discussion.

Some students may not know how to begin sorting the cards. Suggest that they try solving 2–3 systems. Ask them to notice if there's a point in the solving process when they realize how many solutions the system has or what the graphs of the two equations would look like. Encourage to look for similarities in the structure of the equations and see how the structure might be related to the number of solutions. 

Invite groups to share their sorting results and record them. Ask the class if they agree or disagree. If there are disagreements, ask students who disagree to share their reasoning.

Display all the systems—sorted into groups—for all to see and discuss the characteristics of the equations in each group. Ask students questions such as:

• “How can we tell from looking at the equations in cards 2 and 5 that the systems have no solution?” (Possible reasoning:
  • Card 2: The equations are in slope-intercept form. The slope is 2 for both graphs, but the \(y\)-intercept is different, so the lines must be parallel.
  • Card 5: The second equation can be rearranged into \(5x - 20y = \text-60\) and multiplied by \(\frac15\) to give \(x-4y=\text-12\). Both equations now have \(x-4y\) on one side, but that expression is equal to 4 in the first equation and equal to -4 in the second. There are no pair of \(x\) and \(y\) that can make both equations true.)
• “What about the equations in cards 3, 7, and 8? What features might give us a clue that the systems have many solutions?” (Possible reasoning:
  • Card 3: The coefficients and constants in the second equation are 3 times those in the first, so they are equivalent equations.
  • Card 7: The first equation can be rearranged into the same form as the first: \(x -4y = \text-4\). We can then see that the second equation, \(4x-16y=\text-16\), is a multiple of the first equation, so they have all the same solutions.
  • Card 8: The first equation can be rearranged to \(y=\text-25 x - \frac15\). We can then see that it is related to the second equation by a factor of 5, so they are equivalent equations.)

We can reason that all the other systems have one solution by a process of elimination—by noticing that they don’t have the features of systems with many solutions or systems with no solutions.

This optional activity gives students another opportunity to apply what they learned about the features of systems of linear equations with one solution, zero solutions, and many solutions. In earlier activities, students were given systems of equations and were asked to determine the number of solutions. Here, they are given one equation and are to write a second one such that the two equations form a system with one solution, zero solutions, and many solutions.

To answer the first question (a system with one solution), students could write a second equation with randomly chosen parameters. Answering the second and third questions, however, relies on an understanding of what "zero solutions" and "infinitely many solutions" mean and how these conditions are visible in the pair of equations and graphically.

For example, students could reason that in a system with no solutions:

• The two equations have the same variable expressions on one side but different numbers on the other side, and then write a second equation accordingly. For instance, if \(5x-2y\) is equal to 10, it cannot also be equal to 4, so \(5x-2y=10\) and \(5x-2y=4\) would form a system with no solutions.
• The graphs of the equations have the same slope, but they cross the vertical axis at different points. Rewriting \(5x-2y =10\) in the form of \(y=mx+b\) gives \(y=\text-\frac52x + 10\). A second equation with a coefficient of \(\text-\frac52\) for \(x\) but a different constant would have a graph that is parallel to the graph of the first equation, forming a system with no solutions.

Regardless of the form students use to write the second equation, they need to choose the parameters strategically to achieve a system with the desired number of solutions. The work here prompts students to look for and make use of structure (MP7).

Consider keeping students in groups of 2. Encourage students to think quietly about the first question and then discuss their response with their partner, and to do the same with the remaining two questions.

Invite students to share their equations. If possible, use graphing technology to graph each equation students share and to verify that the resulting system indeed has the specified number of solutions. Display the graphs for all to see.

Focus the discussion on how students wrote an equation that would produce a system with no solutions and a system with infinitely many solutions. Highlight strategies that show an understanding of equivalent equations, of the meaning of solutions to equations in two variables, and of the graphical features of these systems.