The Division and Multiplication Properties of Equality

Solve Equations That Require Simplification

Many equations start out more complicated than the ones we have been working with.

With these more complicated equations the first step is to simplify both sides of the equation as much as possible. This usually involves combining like terms or using the distributive property.

Example 2.18

Solve: \(14−23=12y−4y−5y\).

Solution

Begin by simplifying each side of the equation.

		\(14-23 \stackrel{?}{=}-36+12+15\)
Simplify each side.		\(-9=-9 \text{✓}\)
Divide both sides by \(3\) to isolate \(y\).		\(14-23=12 y-4 y-5 y\)
Divide.		\(-9=3 y\)
Check:	\(14-23=12 y-4 y-5 y\)
Substitute \(y=−3\).	\(14-23 \stackrel{?}{=} 12(-3)-4(-3)-5(-3)\)
	\(14-23 \stackrel{?}{=}-36+12+15\)
	\(-9=-9 \text{✓}\)

Try It 2.35

Solve: \(18−27=15c−9c−3c\).

Try It 2.36

Solve: \(18−22=12x−x−4x\).

Example 2.19

Solve: \(−4(a−3)−7=25\).

Solution

Here we will simplify each side of the equation by using the distributive property first.

		\(-4(a-3)-7=25\)
Distribute.		\(-4 a+12-7=25\)
Simplify.		\(-4 a+5=25\)
Simplify.		\(-4 a=20\)
Divide both sides by \(−4\) to isolate \(a\).		\(\frac{-4 a}{-4}=\frac{20}{-4}\)
Divide.		\(a=-5\)
Check:	\(-4(a-3)-7=25\)
Substitute \(a=−5\).	\(-4(-5-3)-7 \stackrel{?}{=} 25\)
	\(-4(-8)-7 \stackrel{?}{=} 25\)
	\(32-7 \stackrel{?}{=} 25\)
	\(25=25 \text{✓}\)

Try It 2.37

Solve: \(−4(q−2)−8=24\).

Try It 2.38

Solve: \(−6(r−2)−12=30\).

Now we have covered all four properties of equality - subtraction, addition, division, and multiplication. We'll list them all together here for easy reference.

Properties of Equality

Subtraction Property of Equality

For any real numbers \(a, b\), and \(c\),
\(\begin{array}{l} \text { if } \quad a=b \\
\text { then } a-c=b-c. \end{array}
\)

Addition Property of Equality

For any real numbers \(a, b\), and \(c\),

\(\begin{array}{l}
\text { if } \quad a=b \\ \text { then } a+c=b+c . \end{array}\).

Division Property of Equality

For any numbers \(a, b\), and \(c\), and \(c \neq 0\),
\(\begin{array}{ll}\text { if } & a=b, \\ \text { then } \frac{a}{c} & =\frac{b}{c} .\end{array}\).

Multiplication Property of Equality

For any numbers \(a, b\), and \(c\),
\(\begin{array}{lrl} \text { if } \quad a & =b \\ \text { then } a c & =b c .
\end{array} \)

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