3.5 Solve equations using integers; the division property of  (Page 3/5)

Solve: $\mathrm{-3}y=63.$

Solution

To isolate $y,$ we need to undo the multiplication.

Divide each side by −3.
Simplify

Check the solution.

	$\mathrm{-3}y=63$
Substitute −21 for y.	$\mathrm{-3}\left(\mathrm{-21}\right)\stackrel{?}{=}63$
	$63=63✓$

Since this is a true statement, $y=\mathrm{-21}$ is the solution to the equation.

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Translate to an equation and solve

In the past several examples, we were given an equation    containing a variable. In the next few examples, we’ll have to first translate word sentences into equations with variables and then we will solve the equations.

Key concepts

• How to determine whether a number is a solution to an equation.
  • Step 1. Substitute the number for the variable in the equation.
  • Step 2. Simplify the expressions on both sides of the equation.
  • Step 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.
• Properties of Equalities

  Subtraction Property of Equality	Addition Property of Equality
  $\text{For any numbers}a,b,c,$ $\text{if}a=b\text{then}a-c=b-c.$	$\text{For any numbers}a,b,c,$ $\text{if}a=b\text{then}a+c=b+c.$

• Division Property of Equality
  • For any numbers $a,b,c,$ and $c\ne 0$
    If $a=b$ , then $\frac{a}{c}=\frac{b}{c}$ .

Practice makes perfect

Determine Whether a Number is a Solution of an Equation

In the following exercises, determine whether each number is a solution of the given equation.

Solve Equations Using the Addition and Subtraction Properties of Equality

In the following exercises, solve for the unknown.

Model the Division Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters and then solve it.