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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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Practice Multi-Step Equations with Like Terms
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Multi-Step Equations with Like Terms

What if you had an equation in which the same variable appeared twice, like 2(x4)+4x=23\begin{align*}2(x - 4) + 4x =-23\end{align*}? How could you simplify the equation so that the variable appears only once in order to solve for it? After completing this Concept, you'll be able to combine like terms to solve two-step equations like this one.

### Guidance

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*}

#### Example A

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

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

#### Example B

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*}.

#### Example C

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

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*}.

Watch this video for help with the Examples above.

### Vocabulary

• Terms with the same variable in them (or no variable in them) are like terms. Combine like terms (adding or subtracting them from each other) to simplify the expression and solve for the unknown.

### Guided Practice

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

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*}.

### Explore More

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*}