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Solving Equations That Require Simplification

Solving 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

Solve: \(14-23=12y-4y-5y.\)

Solution

Begin by simplifying each side of the equation.

.
Simplify each side. .
Divide both sides by \(3\) to isolate \(y\). .
Divide. .
Check: .
Substitute \(y=-3\). .
.
.

Example

Solve: \(-4\left(a-3\right)-7=25.\)

Solution

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

.
Distribute. .
Simplify. .
Simplify. .
Divide both sides by \(-4\) to isolate \(a\). .
Divide. .
Check: .
Substitute \(a=-5\). .
.
.
.

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

\(\begin{array}{ccc}\mathbf{\text{Subtraction Property of Equality}}\hfill & & \mathbf{\text{Addition Property of Equality}}\hfill \\ \text{For any real numbers}\phantom{\rule{0.2em}{0ex}}a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c,\hfill & & \text{For any real numbers}\phantom{\rule{0.2em}{0ex}}a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c,\hfill \\ \phantom{\rule{1em}{0ex}}\begin{array}{cccc}\text{if}\hfill & \hfill a& =\hfill & b,\hfill \\ \text{then}\hfill & \hfill a-c& =\hfill & b-c.\hfill \end{array}\hfill & & \phantom{\rule{1em}{0ex}}\begin{array}{cccc}\text{if}\hfill & \hfill a& =\hfill & b,\hfill \\ \text{then}\hfill & \hfill a+c& =\hfill & b+c.\hfill \end{array}\hfill \\ \mathbf{\text{Division Property of Equality}}\hfill & & \mathbf{\text{Multiplication Property of Equality}}\hfill \\ \text{For any numbers}\phantom{\rule{0.2em}{0ex}}a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c,\text{and}\phantom{\rule{0.2em}{0ex}}c\ne 0,\hfill & & \text{For any numbers}\phantom{\rule{0.2em}{0ex}}a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c,\hfill \\ \phantom{\rule{1em}{0ex}}\begin{array}{cccc}\text{if}\hfill & \hfill a& =\hfill & b,\hfill \\ \text{then}\hfill & \hfill \frac{a}{c}& =\hfill & \frac{b}{c}.\hfill \end{array}\hfill & & \phantom{\rule{1em}{0ex}}\begin{array}{cccc}\text{if}\hfill & \hfill a& =\hfill & b,\hfill \\ \text{then}\hfill & \hfill ac& =\hfill & bc.\hfill \end{array}\hfill \end{array}\)

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

This lesson is part of:

Solving Linear Equations II

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