Systems of Linear Equations and Their Solutions

10 min

Teacher Prep
Required Preparation
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if students each have their own device. (Desmos is available under Math Tools.)

Narrative

This Warm-up reminds students about the fact that some systems of linear equations have many solutions, 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 5, so students choose a pair of values whose sum is 5. 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=5−xy = 5-x. Substituting 5−x5-x for yy in the second equation gives 4x=20−4(5−x)4x = 20 - 4(5-x), or 4x=20−20+4x4x = 20-20+4x, or 4x=4x4x=4x. This equation is true no matter what xx is.
  • Solving by elimination: If we multiply the first equation by 4, rearrange the second equation to 4x+4y=204x+4y=20, and then subtract the second equation from the first, the result is 4y=4y4y=4y. Subtracting 4y4y from each side gives 0=00=0, which is true regardless of what xx or yy is.
  • Reasoning about equivalent equations: If we rearrange the second equation so that the variables are on the same side 4x+4y=204x+4y=20, we can see that this equation is a multiple of the first and that the two equations are equivalent. This means they have the exact same solution set, which contains infinite possible pairs of xx and yy.

Identify students who use 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).

Launch

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 to 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 5 is also a solution to the system, without having to do calculations for each pair?" 

Student Task

Andre is trying to solve this system of equations: {x+y=54x=20−4y\begin {cases} x + y = 5\\ 4x = 20 - 4y \end{cases}

Looking at the first equation, he thought, "The solution to the system is a pair of numbers that add up to 5. I wonder which two numbers they are." 

  1. Choose any two numbers that add up to 5. Let the first one be the xx-value and the second one be the yy-value.
  2. The pair of values you chose is a solution to the first equation. Check if it is also a solution to the second equation. Then, pause for a brief discussion with your group.
  3. How many solutions does the system have? Use what you know about equations or about solving systems to show that you are right.

Sample Response

  1. Sample responses: 6 and -1, 0 and 5, 12\frac{1}{2} and 4124\frac{1}{2}
  2. Yes, it is also a solution to the second equation. Sample reasoning for 6 and -1: Substituting 6 and -1 into the second equation gives  4(6)=20−4(-1)4(6) = 20 - 4(\text{-}1), or 24=20+424 = 20 + 4, which is a true equation.
  3. Infinitely many solutions. See the Activity Narrative for some possible explanations.
Activity Synthesis (Teacher Notes)

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 5 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 Activity 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 xx and yy values that make one equation true also make the other equation true, giving an infinite number of solutions.

Standards
Building On
  • 8.EE.8·Analyze and solve pairs of simultaneous linear equations.
  • 8.EE.C.8·Analyze and solve pairs of simultaneous linear equations.
Addressing
  • A-REI.6·Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
  • A-REI.6·Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
  • A-REI.6·Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
  • HSA-REI.C.6·Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

10 min

15 min