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3.5 Solve equations using integers; the division property of

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By the end of this section, you will be able to:
  • Determine whether an integer is a solution of an equation
  • Solve equations with integers using the Addition and Subtraction Properties of Equality
  • Model the Division Property of Equality
  • Solve equations using the Division Property of Equality
  • Translate to an equation and solve

Before you get started, take this readiness quiz.

  1. Evaluate x + 4 when x = −4 .
    If you missed this problem, review Evaluate, Simplify and Translate Expressions .
  2. Solve: y − 6 = 10 .
    If you missed this problem, review Solve Equations with the Subtraction and Addition Properties of Equality .
  3. Translate into an algebraic expression 5 less than x .
    If you missed this problem, review Subtract Whole Numbers .

Determine whether a number is a solution of an equation

In Solve Equations with the Subtraction and Addition Properties of Equality , we saw that a solution of an equation is a value of a variable that makes a true statement when substituted into that equation. In that section, we found solutions that were whole numbers. Now that we’ve worked with integers, we’ll find integer solutions to equations.

The steps we take to determine whether a number is a solution to an equation are the same whether the solution is a whole number or an integer.

How to determine whether a number is a solution to an equation.

  1. Substitute the number for the variable in the equation.
  2. Simplify the expressions on both sides of the equation.
  3. Determine whether the resulting equation is true.
    • If it is true, the number is a solution.
    • If it is not true, the number is not a solution.

Determine whether each of the following is a solution of 2 x − 5 = −13 :

  1. ⓐ x = 4
  2. ⓑ x = −4
  3. ⓒ x = −9 .

Solution

ⓐ Substitute 4 for x in the equation to determine if it is true.
.
. .
Multiply. .
Subtract. .

Since x = 4 does not result in a true equation, 4 is not a solution to the equation.

ⓑ Substitute −4 for x in the equation to determine if it is true. .
. .
Multiply. .
Subtract. .

Since x = −4 results in a true equation, −4 is a solution to the equation.

ⓒ Substitute −9 for x in the equation to determine if it is true.
.
Substitute −9 for x. .
Multiply. .
Subtract. .

Since x = −9 does not result in a true equation, −9 is not a solution to the equation.

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Determine whether each of the following is a solution of 2 x − 8 = −14 :

  1. ⓐ x = −11
  2. ⓑ x = 11
  3. ⓒ x = −3

  1. ⓐ no
  2. ⓑ no
  3. ⓒ yes
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Determine whether each of the following is a solution of 2 y + 3 = −11 :

  1. ⓐ y = 4
  2. ⓑ y = −4
  3. ⓒ y = −7

  1. ⓐ no
  2. ⓑ no
  3. ⓒ yes
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Solve equations with integers using the addition and subtraction properties of equality

In Solve Equations with the Subtraction and Addition Properties of Equality , we solved equations similar to the two shown here using the Subtraction and Addition Properties of Equality. Now we can use them again with integers.

This figure has two columns. The first column has the equation x plus 4 equals 12. Underneath there is x plus 4 minus 4 equals 12 minus 4. Under this there is x equals 8. The second column has the equation y minus 5 equals 9. Underneath there is the equation y minus 5 plus 5 equals 9 plus 5. Under this there is y equals 14.

When you add or subtract the same quantity from both sides of an equation, you still have equality.

Properties of equalities

Subtraction Property of Equality Addition Property of Equality
For any numbers a , b , c ,
if a = b then a − c = b − c . 		For any numbers a , b , c ,
if a = b then a + c = b + c .

Solve: y + 9 = 5 .

Solution

.
Subtract 9 from each side to undo the addition. .
Simplify. .

Check the result by substituting −4 into the original equation.

y + 9 = 5
Substitute −4 for y −4 + 9 = ? 5
5 = 5 ✓

Since y = −4 makes y + 9 = 5 a true statement, we found the solution to this equation.

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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