The Algebra Teacher\'s Guide to Reteaching Essential Concepts and Skills

Teaching Notes 3.7: Solving Equations with Variables

on Both Sides, Including Identities and Equations That Have

No Solution

Solving equations that have variables on both sides presents problems for many students. Even if

they are able to solve equations that have one solution, they are often confused by equations

that are true for all real numbers or that have no solution.

1. Explain that some equations have only one solution, some are true for all real numbers, and
   some have no solution. Depending on the abilities of your students, you may find it helpful
   to review 3.6: ‘‘Solving Equations with Variables on Both Sides.’’


2. Review the information and examples on the worksheet with your students. Explain each
   step fully. Make sure that your students understand the meaning of ‘‘no solution,’’ ‘‘true
   statement,’’ and ‘‘identity.’’ Emphasize the following points:
   
   
   • If an equation is equivalent to a true statement, then the equation is true for all real
     numbers.
   
   
   • If an equation is equivalent to a false statement, then the equation has no solution.
   
   
   
   

EXTRA HELP:

Double-check final equations to determine if all real numbers are a solution or if there is no

solution.

ANSWER KEY:

(1)x= 5 (2)No solution (3)Identity (4)No solution (5)y= 0 (6)No solution

(7)Identity (8)No solution

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(Challenge)Leah’s answer is incorrect. She ignored the parentheses and treated 4(x+2) as

4 x+2. She subtracted 3xand then 2 from both sides of the equation. The correct answer is

x=−5.

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100 THE ALGEBRA TEACHER’S GUIDE