Solving Systems by Elimination (Part 2)

5 min

Narrative

In this Warm-up, students reason about whether and when the sums of equations are true. The work here prepares students for the next activity, in which they begin to think about why the values that simultaneously satisfy two equations in a system also satisfy the equation that is a sum of those two equations.

Student Task

Here is a true equation: 50+1=5150 + 1 = 51.

  1. Perform each of the following operations and answer these questions: What does each resulting equation look like? Is it still a true equation?

    1. Add 12 to each side of the equation.
    2. Add 10+210 + 2 to the left side of the equation and 12 to the right side.
    3. Add the equation 4+3=74 + 3 = 7 to the equation 50+1=5150 + 1 = 51.
  2. Write a new equation that, when added to 50+1=5150 +1 = 51, gives a sum that is also a true equation.
  3. Write a new equation that, when added to 50+1=5150 +1 = 51, gives a sum that is a false equation.

Sample Response

    1. 50+1+12=51+1250 + 1 + 12 = 51 + 12, or 63=6363 = 63 (or equivalent). Yes, it is still a true equation.
    2. 50+1+10+2=51+1250 + 1 + 10 + 2 = 51 + 12, or 63=6363 = 63 (or equivalent). Yes, it is still a true equation.
    3. 50+1+4+3=51+750 + 1 + 4 + 3 = 51 + 7, or 54+4=5854 + 4 = 58, or 58=5858=58 (or equivalent). Yes, it is still a true equation.
  2. Sample responses:
    • 8+2=108 + 2 = 10. Added to the original equation gives: 50+1+8+2=51+1050 + 1 + 8 + 2 = 51 + 10, or 51+10=6151 + 10 = 61, or 61=6161=61
    • a+a=2aa + a = 2a. Added to the original equation gives: 50+1+a+a=51+2a50 + 1 + a + a = 51 + 2a, or 51+2a=51+2a51 +2a = 51 + 2a
  3. Sample responses:
    • 3+2=73 + 2 = 7. Added to the original equation gives: 50+1+3+2=5850 + 1 + 3 + 2 = 58, or 56=5856 = 58, which is false.
    • a+2=aa +2 =a. Added to the original equation gives: 50+1+a+2=51+a50 + 1 + a + 2 = 51 + a, or 53+a=51+a53 + a = 51 + a, which is false.
Activity Synthesis (Teacher Notes)

Ask students to share their responses to the first three questions. Discuss questions such as:

  • "Why do you think the resulting equations remain true even after we add numbers or expressions that look different to each side?" (The numbers being added to the two sides are always equal amounts, even though they are written in a different form.)
  • "Suppose we subtract 12 from the left side of 50+1=5150+1=51 and subtract 5+75+7 from the right side. Would the resulting equation still be true?" (Yes. 50+1−1250+1-12 is 39 and 51−(5+7)51 - (5+7) is also 39. The same amount, 12, is subtracted from both sides.)
  • "Here is another equation: 50+1=6050 + 1 = 60. Suppose we subtract 12 from the left side of that equation and 5+75+7 from the right side. Would the resulting equation be true? Why or why not?" (No. The given equation is a false statement. Subtracting the same amount from both sides will keep it false.)

Invite students to share their equations for the last two questions. Display the equations for all to see. If no one shares an equation that uses a variable, give an example or two (as shown in the Student Responses).

Make sure students understand that adding (or subtracting) the same amount to (or from) each side of a true equation keeps the two sides equal, resulting in an equation that is also true. Adding (or subtracting) different amounts to (or from) each side of a true equation, however, makes the two sides unequal and thus produces a false equation.

Standards
Addressing
  • F-BF.1.b·Combine standard function types using arithmetic operations. <em>For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.</em>
  • F-BF.1.b·Combine standard function types using arithmetic operations. <em>For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.</em>
  • F-BF.1.b·Combine standard function types using arithmetic operations. <em>For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.</em>
  • F-BF.1.b·Combine standard function types using arithmetic operations. <em>For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.</em>
  • F-BF.1.b·Combine standard function types using arithmetic operations. <em>For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.</em>
  • F-BF.1.b·Combine standard function types using arithmetic operations. <em>For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.</em>
  • HSF-BF.A.1.b·Combine standard function types using arithmetic operations. <span>For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.</span>
Building Toward
  • A-REI.5·Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
  • A-REI.5·Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
  • A-REI.5·Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
  • HSA-REI.C.5·Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

20 min

10 min