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# Multi-Step Equations with Like Terms

## Add and subtract like terms as a step to solve equations.

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Multi-Step Equations with Like Terms

### 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 x\begin{align*}x\end{align*} on both sides, we know that in order to solve it, we need to get all the x\begin{align*}x-\end{align*}terms on one side of the equation. This is called combining like terms. The terms with an x\begin{align*}x\end{align*} in them are 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
4x,10x,3.5x,\begin{align*}4x, 10x, -3.5x,\end{align*} and x12\begin{align*}\frac{x}{12}\end{align*} 3x\begin{align*}3x\end{align*} and 3y\begin{align*}3y\end{align*}
3y,0.000001y,\begin{align*}3y, 0.000001y,\end{align*} and y\begin{align*}y\end{align*} 4xy\begin{align*}4xy\end{align*} and 4x\begin{align*}4x\end{align*}
xy,6xy,\begin{align*}xy, 6xy,\end{align*} and 2.39xy\begin{align*}2.39xy \end{align*} 0.5x\begin{align*}0.5x\end{align*} and 0.5\begin{align*}0.5 \end{align*}

#### Using the Distributive Property of Multiplication

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

3x+4x0.03xy0.01xyy+16y+5y5z+2z7z=(3+4)x=7x=(0.030.01)xy=0.02xy=(1+16+5)y=10y=(5+27)z=0z=0\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 (x+5)(2x3)=6\begin{align*}(x + 5) - (2x - 3)=6\end{align*}.

There are two like terms: the x\begin{align*}x\end{align*} and the 2x\begin{align*}-2x\end{align*} (don’t forget that the negative sign applies to everything in the parentheses). So we need to get those terms together. The associative and distributive properties let us rewrite the equation as x+52x+3=6\begin{align*} x + 5 - 2x + 3 = 6\end{align*}, and then the commutative property lets us switch around the terms to get x2x+5+3=6\begin{align*}x - 2x + 5 + 3 = 6\end{align*}, or (x2x)+(5+3)=6\begin{align*}(x - 2x) + (5 + 3) = 6\end{align*}.

(x2x)\begin{align*}(x - 2x)\end{align*} is the same as (12)x\begin{align*}(1 - 2)x\end{align*}, or x\begin{align*}-x\end{align*}, so our equation becomes x+8=6\begin{align*}-x + 8 = 6\end{align*}

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

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

2. Solve x2x3=6\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.

3x62x6=6\begin{align*}\frac{3x}{6} - \frac{2x}{6} = 6\end{align*}

Then we subtract the fractions to get x6=6\begin{align*}\frac{x}{6} = 6\end{align*}.

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

### Example

#### Example 1

Solve 2x53x2=11\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.

4x1015x10=11\begin{align*}\frac{4x}{10} - \frac{15x}{10} = 11\end{align*}

Then we subtract the fractions to get 11x10=11\begin{align*}-\frac{11x}{10} = 11\end{align*}.

Finally we multiply both sides by 1011:\begin{align*}-\frac{10}{11} :\end{align*}

11x101011=111011\begin{align*}-\frac{11x}{10}\cdot -\frac{10}{11} = 11 \cdot -\frac{10}{11} \end{align*}

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

### Review

Solve the following equations for the unknown variable.

1. 1.3x0.7x=12\begin{align*}1.3x - 0.7x = 12\end{align*}
2. 10a2(a+5)=14\begin{align*}-10a - 2(a+5) = 14\end{align*}
3. 5(2y3y)=20\begin{align*}5(2y-3y) = -20\end{align*}
4. 23x15x=1415\begin{align*}\frac{2}{3}x - \frac{1}{5}x = \frac{14}{15}\end{align*}
5. 5x(3x+2)=1\begin{align*}5x - (3x + 2) = 1 \end{align*}
6. s3s8=56\begin{align*}s - \frac{3s}{8} = \frac{5}{6}\end{align*}
7. 10(y+5y)=10\begin{align*}10(y + 5y) = 10\end{align*}
8. 2.3x+2(0.75x3.5)=7.5\begin{align*}2.3x+2(0.75x-3.5) = 7.5\end{align*}
9. 3(x+2)+5(2x)=32\begin{align*}3(x+2)+5(2-x)=-32\end{align*}
10. 6x+2(5x2)=12\begin{align*}6x + 2(5x - 2) = 12 \end{align*}

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