More Solutions to Linear Equations

5 min

Teacher Prep
Setup
Encourage students to not pick 0 for xx each time.

Narrative

The purpose of this Warm-up is for students to practice solving an equation for an unknown value while thinking about a coordinate pair, (x,y)(x,y), that makes the equation true. While the steps to solve the equation are the same regardless of which value of xx students choose, there are strategic choices that can make solving the resulting equation simpler.

Launch

Give students 2–3 minutes of quiet work time followed by a whole-class discussion. If necessary, encourage students to not pick 0 for xx each time.

Engagement: Develop Effort and Persistence. Provide tools to facilitate information processing or computation, enabling students to focus on key mathematical ideas. For example, allow students to use a calculator to support their reasoning.
Supports accessibility for: Memory, Conceptual Processing

Student Task

For each equation choose a value for xx and then find the corresponding yy-value that makes that equation true.

  1. 6x=7y6x=7y
  2. 5x+3y=95x+3y=9
  3. y+513x=7y+5-\frac13 x=7

Sample Response

Sample responses:

  1. x=7,y=6x = 7, y = 6
  2. x=3,y=-2x = 3, y = \text-2
  3. x=3,y=3x = 3, y = 3
Activity Synthesis (Teacher Notes)

The goal of this discussion is to reinforce the idea that the solutions to a given equation will all lie on the same line, and that line represents the set of all possible solutions to the equation. Begin by collecting the pairs of xx’s and yy’s students calculated and graphing them on a coordinate plane. It may be useful to graph each set of points in a different color. Here are some questions for discussion: 

  • “How did you pick your xx-values?” (For the first problem, choosing xx to be a multiple of 7 makes yy an integer. For the last problem, picking xx to be a multiple of 3 makes yy an integer.)

  • “What do you notice about all the points?” (The points collected for each equation form a different line.)

Standards
Addressing
  • 8.EE.C·Analyze and solve linear equations and pairs of simultaneous linear equations.
  • 8.EE.C·Analyze and solve linear equations and pairs of simultaneous linear equations.

15 min

15 min