### Multi-Step Equations with Like Terms

When we look at a linear equation we see two kinds of terms: those that contain the unknown variable, and those that don’t. When we look at an equation that has an \begin{align*}x\end{align*}**combining like terms**. The terms with an \begin{align*}x\end{align*}**like terms** because they contain the same variable (or, as you will see in later chapters, the same combination of variables).

Like Terms |
Unlike Terms |
---|---|

\begin{align*}4x, 10x, -3.5x,\end{align*} |
\begin{align*}3x\end{align*} |

\begin{align*}3y, 0.000001y,\end{align*} |
\begin{align*}4xy\end{align*} |

\begin{align*}xy, 6xy,\end{align*} |
\begin{align*}0.5x\end{align*} |

#### Using the Distributive Property of Multiplication

To add or subtract like terms, we can use the Distributive Property of Multiplication.

\begin{align*}3x + 4x &= (3 + 4)x = 7x \\
0.03xy - 0.01xy &= (0.03 - 0.01)xy = 0.02xy\\
-y + 16y + 5y &= (-1 + 16 + 5)y = 10y\\
5z + 2z - 7z &= (5 + 2 - 7)z = 0z = 0\end{align*}

To solve an equation with two or more like terms, we need to combine the terms first.

#### Solving for Unknown Values

1. Solve \begin{align*}(x + 5) - (2x - 3)=6\end{align*}

There are two like terms: the \begin{align*}x\end{align*}

\begin{align*}(x - 2x)\end{align*}

Subtracting 8 from both sides gives us \begin{align*}-x = -2 \end{align*}

And finally, multiplying both sides by -1 gives us \begin{align*}x = 2 \end{align*}

2. Solve \begin{align*}\frac{x}{2} - \frac{x}{3} = 6 \end{align*}

This problem requires us to deal with fractions. We need to write all the terms on the left over a common denominator of six.

\begin{align*}\frac{3x}{6} - \frac{2x}{6} = 6\end{align*}

Then we subtract the fractions to get \begin{align*}\frac{x}{6} = 6\end{align*}

Finally we multiply both sides by 6 to get \begin{align*}x = 36\end{align*}

### Example

#### Example 1

Solve \begin{align*}\frac{2x}{5} - \frac{3x}{2} = 11 \end{align*}

This problem requires us to deal with fractions. We need to write all the terms on the left over a common denominator of ten.

\begin{align*}\frac{4x}{10} - \frac{15x}{10} = 11\end{align*}

Then we subtract the fractions to get \begin{align*}-\frac{11x}{10} = 11\end{align*}

Finally we multiply both sides by \begin{align*}-\frac{10}{11} :\end{align*}

\begin{align*}-\frac{11x}{10}\cdot -\frac{10}{11} = 11 \cdot -\frac{10}{11} \end{align*}

to get \begin{align*}x = -10\end{align*}

### Review

Solve the following equations for the unknown variable.

- \begin{align*}1.3x - 0.7x = 12\end{align*}
1.3x−0.7x=12 - \begin{align*}-10a - 2(a+5) = 14\end{align*}
−10a−2(a+5)=14 - \begin{align*}5(2y-3y) = -20\end{align*}
5(2y−3y)=−20 - \begin{align*}\frac{2}{3}x - \frac{1}{5}x = \frac{14}{15}\end{align*}
23x−15x=1415 - \begin{align*}5x - (3x + 2) = 1 \end{align*}
5x−(3x+2)=1 - \begin{align*}s - \frac{3s}{8} = \frac{5}{6}\end{align*}
s−3s8=56 - \begin{align*}10(y + 5y) = 10\end{align*}
10(y+5y)=10 - \begin{align*}2.3x+2(0.75x-3.5) = 7.5\end{align*}
2.3x+2(0.75x−3.5)=7.5 - \begin{align*}3(x+2)+5(2-x)=-32\end{align*}
3(x+2)+5(2−x)=−32 - \begin{align*}6x + 2(5x - 2) = 12 \end{align*}
6x+2(5x−2)=12

### Review (Answers)

To view the Review answers, open this PDF file and look for section 3.5.