In the previous sections, you used two different models to solve equations: algebra tiles and cups and counters. When you were given an equation, you used models as well as inverse operations to determine the solution to the equation.

In this section, you will work backwards. If you have a possible solution, you will use substitution to determine if that value really is a solution to a given equation.

Source: Mother cat, rones, Open Clipart

Use the interactive below to see how three students, Emily, Ly, and Dashon determined whether or not x = 4.5 is a solution to each equation. Click on each student in the picture below to see how they approached one of the three equations. In this interactive, you will need to know that the symbol, , an equal sign with a question mark above it means "might be equal to."

Use the interactive to answer the questions that follow.

• Which equations have a solution of x = 4.5?

  Interactive popup. Assistance may be required.

  Check Your Answer

  Only two equations: 3x = 13.5 and x + 9.4 = 13.9.

• In this problem, how did each of the three students test the value of x to see if it was a solution to their equation?

  Interactive popup. Assistance may be required.

  Check Your Answer

  Each student used substitution. They substituted 4.5 for the value of x in each equation, and then simplified the left side of the equation to see if it generated the same value as the right side of the equation.

• After the expression on the left side of the equation is simplified, how does it compare to the right side of the equation if the value of x is a solution to the equation?

  Interactive popup. Assistance may be required.

  Check Your Answer

  If the value of x is a solution to the equation, then the left side of the equation has the same value as the right side of the equation.

Practice

1. For which of the following equations is -5 a solution?
   1. -3n = 15
   2. 77 + x = 82
   3. j − 94 = -99
   Interactive popup. Assistance may be required.

   Need a hint?

   Substitute -5 for the variable in each equation. Determine which substitution generates a true statement (i.e., a statement that is true, such as 5 = 5).
   Interactive popup. Assistance may be required.

   Check Your Answer

   I and III only
   I. -3n = 15
   -3(-5) 15
   15 = 15
   
   II. 77 + x = 82
   77 + (-5) 82
   72 ≠ 82
   
   III. j − 94 = -99
   (-5) − 94 -99
   -99 = -99
2. Marcus and Antoine both drew a random number from a bag. Marcus drew 1.5 and Antoine drew 2.5. Which student drew a number that is a solution to the equation 8x = 12?

   Interactive popup. Assistance may be required.

   Need a hint?

   Substitute -5 for the variable in each equation. Determine which substitution generates a true statement (i.e., a statement that is true, such as 5 = 5).
   Interactive popup. Assistance may be required.

   Check Your Answer

   Marcus. 8x = 12 8(1.5) 12 12 = 12

   8x = 12 8(2.5) 12 20 ≠ 12

   
3. The length of a rectangle is 2x centimeters and the width is 4 centimeters. The area of the rectangle is 32 square centimeters. Olivia claims the value of x is 8. Is Olivia correct? Why or why not?

   Interactive popup. Assistance may be required.

   Need a hint?

   The area of a rectangle is found using the formula A = lw. Use this formula to write an equation in terms of x, and then determine if x = 8 is the solution to this equation.
   Interactive popup. Assistance may be required.

   Check Your Answer

   No, Olivia is not correct.
   The length, width, and area of the rectangle are related by the equation 2x(4) = 32, or 8x = 32. Determine whether or not Olivia’s solution of x = 8 works in this equation.
   8x = 32
   8(8) 32
   64 ≠ 32