Solving Equations with Variables on Both Sides of the Equation

GOAL: I can solve problems that have variables on both sides of the equation.

Text reference: Chapter 2, section 4, pages: 102 ─ 108 
Illinois learning standard: 8.C.3 
NCTM standard: XX─XX 
Lesson Vocabulary 

Identity Equation

An identity equation is a equation that when solved, has the exact same value on both sides of the equation. For example, if we simplify the following:
2x + 5 = 5x + 5 ─ 3x
When we combine the 5x and (─3x) on the right side of the equation, we get:
2x + 5 = 2x + 5
The exact same binomial is on the left side of the equation as on the right side of the equation. This is an identity equation.
ANY value that is substituted in place of the variable "x" will make this a true equation. That means that every value in the universe of numbers is an answer to this problem. And since there is NOT just one unique answer to this equation, it is called an identity equation.

Internet Video Tutorials    Topic  Problem  Time (min: sec)  Solving Equations with Variables on Both Sides of the Equation
 20 ─ 7y = 6y ─ 6
 4:33  Solving Equations with Variables on Both Sides of the Equation  x ─ 8 = (x/3) + (1/6)
 5: 46


Strategies and Content Practice 
Notes 
Begin
(prepare) 
Check for prior knowledge. Discuss and review solving multistep problems. Discuss and review using the distributive property.


Engage 
Have students read pages 102 ─ 106. Read aloud same with students. Demonstrate the identity equation example given in the definition section.
Equations with variables on both sides an extension of solving multistep equations. The techniques used in solving multistep equations generally apply to only one side of the equation. In this lesson, the techniques for solving multistep equations are applied to both sides of the equation. In short, solving problems with variables on both sides of the equation simply requires an additional simplification step or two.
Three outcomes or "cases" will occur when solving problems with variables on both sides of the equal sign. These cases are explained below.
Case #1 In case #1, the equation can be solved for a numeric value. This is normally what you might expect to happen.
Case #2 and Case #3 In case #2 and case #3, the variables on both sides of the equal sign turn out to be identical. When variables on both sides of the equal sign are identical, they cancel each other out and disappear from the equation. What is left over after the variables are gone?
Case #2 In case #2, the variables on both sides of the equal sign were identical and they cancelled each other out.
7x + 8 = 7x + (─4) (7x cancels from both sides leaving...) 8 ≠ (─4) or "NO SOLUTION"
There are now two numbers on both sides of the equal sign. If the leftover numbers on the left and right sides of the equation are not identical (different values), then we call this case a "NO SOLUTION" case.
Case #3 In case #3, the variables on both sides of the equal sign were identical and they cancelled each other out.
─2x + 5 = ─2x + 5
(─2x cancels from both sides leaving...)
5 = 5 or "IDENTITY"
There are now two numbers on both sides of the equal sign. If the
leftover numbers on the left and right sides of the equation are not
identical (different values), then we call this case a "IDENTITY" case.
Show demonstration videos for Lesson 2─4 (above). 
Try this online Algebra Tiles App to practice solving problems with variables on both sides of the equal sign. 
Assess
(formative) 
After example problems and videos, have students try to solve the Sample Problems below. After students have had time to complete them, click on the Solutions link and go through the stepbystep process for solving each problem. Repeat if necessary with additional example problems in the book and reassess. 

Apply 
Click HERE to download the Practice Assignment.

Depending on class conditions, you may allow students to work with each other.

Assess (formative) 
Click HERE to download the Answers to the Practice Assignment.  Depending on level of understanding, you may accept this preassessment grade and allow students who have mastered these concepts to skip the formal assessment exercise.  Apply  Click HERE to download the Graded Assignment.
 
Sample Problems 
Sample Solutions 
5x + 2 = 2x + 14

Click Here for stepbystep solution. 
3x + 4 = 5x ─ 10

Click Here for stepbystep solution. 
5(x ─ 4) = 7(2x + 1) 
Click Here for stepbystep solution. 
2x + 3 = ½(4x + 3) 
Click Here for stepbystep solution. 
