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# Multi-Step Equations

## Maintain balance of an equation throughout all steps needed to solve.

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Solving Multi-Step Equations

Cell-U-Lar cell phone company includes 500 texts in their $79/month fee and then charges$0.10 for each additional text. We-B-Mobile cell phone company also includes 500 texts in their $59/month fee and then charges$0.25 for each additional text. At how many texts, over 500, will the two bills be the same?

### Multi-Step Equations

The types of equations in this concept involve at least three steps. Keep in mind that the last two steps when solving a linear equation will always be the same: add or subtract the number that is on the same side of the equals sign as the variable, then multiply or divide by the number with the variable.

Let's solve the following equations.

1. Solve 3(x5)+4=10\begin{align*}3(x-5)+4=10\end{align*}.

When solving more complicated equations, start with one side and simplify as much as you can. The left side of this equation looks more complicated, so let’s simplify it by using the Distributive Property and combining like terms.

3(x5)+43x15+43x11=10=10=10\begin{align*}3(x-5)+4 &= 10\\ 3x-15+4 &= 10\\ 3x-11 &= 10\end{align*}

Now, this equation should look familiar. Continue to solve.

3x11=10  +11 +113x3=213  x=7\begin{align*}& \ 3x-\bcancel{11}=10\\ & \underline{\quad \ \ + \bcancel{11} \ + 11 \; \;}\\ & \qquad \frac{\bcancel{3}x}{\bcancel{3}}=\frac{21}{3}\\ & \qquad \ \ x=7\end{align*}

Check your answer: 3(75)+4=32+4=6+4=10\begin{align*}3(7-5)+4=3 \cdot 2+4=6+4=10 \end{align*}

1. Solve 8x17=4x+23\begin{align*}8x-17=4x+23\end{align*}.

This equation has x\begin{align*}x\end{align*} on both sides of the equals sign. Therefore, we need to move one of the x\begin{align*}x\end{align*} terms to the other side of the equation. It does not matter which x\begin{align*}x\end{align*} term you move. We will move the 4x\begin{align*}4x\end{align*} to the other side so that, when combined, the x\begin{align*}x\end{align*} term is positive.

8x17=4x+234x  4x4x17=23+17 +174x4=404  x=10\begin{align*}& 8x-17=\bcancel{4x}+23\\ & \underline{-4x \quad \ \ -\bcancel{4x} \; \; \; \; \; \; \; \;}\\ & 4x-\bcancel{17}=23\\ & \underline{\quad + \bcancel{17} \ +17 \; \; \; \; \; \; \; \; \;}\\ & \qquad \frac{\bcancel{4}x}{\bcancel{4}}=\frac{40}{4}\\ & \qquad \ \ x=10\end{align*}

8(10)17801763=4(10)+23=40+23=63\begin{align*}8(10)-17 &= 4(10)+23\\ 80-17 &= 40+23 \\ 63 &= 63\end{align*}

1. Solve 2(3x1)+2x=5(2x3)\begin{align*}2(3x-1)+2x=5-(2x-3)\end{align*}.

This equation combines what was present in the previous two problems. First, use the Distributive Property.

2(3x1)+2x6x2+2x=5(2x3)=52x+3\begin{align*}2(3x-1)+2x &= 5-(2x-3)\\ 6x-2+2x &= 5-2x+3\end{align*}

Don’t forget to distribute the negative sign in front of the second set of parenthesis. Treat it like distributing a –1. Now, combine like terms and solve the equation.

8x2=82x+2x+2+2+2x 10x10=1010 x=1\begin{align*}& \ \ \ 8x-\bcancel{2}=8-\bcancel{2x}\\ & \underline{+2x+\bcancel{2} +2+\bcancel{2x}}\\ & \quad \ \frac{\bcancel{10}x}{\bcancel{10}}=\frac{10}{10}\\ & \qquad \ x=1\end{align*}

2(3(1)1)+2(1)22+24+2=5(2(1)3)=5(1)=6\begin{align*}2(3(1)-1)+2(1) &= 5-(2(1)-3)\\ 2 \cdot 2+2 &= 5-(-1) \\ 4+2 &= 6\end{align*}

### Examples

#### Example 1

Earlier, you were asked to find when the two bills will be the same for texts over 500.

Let's write an expression for the total cost associated with each cell phone company.

Cell-U-Lar: 79+0.1t\begin{align*}79+0.1t\end{align*}

We-B-Mobile: 59+0.25t\begin{align*}59+0.25t\end{align*}

Where t is the number of texts over 500. Set the two expressions equal to each other to see when the plans are equal.

79+0.1t20133.3¯=59+0.25t=0.15t=t\begin{align*}79+0.1t&=59+0.25t\\ 20&=0.15t\\ 133. \bar 3 &=t\end{align*}

At 133 additional texts, the bills would be just about the same, but don't forget that each company includes 500 texts at no additional cost. So, you would have to go just over 633 texts to make the two plans equal. If you text less than that, the plan from We-B-Mobile is a better deal. More than 633, then Cell-U-Lar makes more sense.

#### Example 2

Solve the following equation: 34+23x=2x+56\begin{align*}\frac{3}{4}+\frac{2}{3}x=2x+\frac{5}{6}\end{align*}. Check your answer.

Use the LCD method: multiply every term by the LCD of 4, 3, and 6.

12(34+23x=2x+56)9+8x=24x+10\begin{align*}& 12 \left(\frac{3}{4}+\frac{2}{3}x = 2x+\frac{5}{6}\right)\\ & \qquad 9+8x = 24x+10\end{align*}

Now, combine like terms. Follow the steps as in problems #2 and #3 above.

9+8x=24x+10108x  8x10116=16x16116=x\begin{align*}& \quad 9+\bcancel{8x}=24x+\bcancel{10}\\ & \underline{-10-\bcancel{8x} \ \ -8x-\bcancel{10} \; \;}\\ & \qquad \frac{-1}{16}=\frac{\bcancel{16}x}{\bcancel{16}}\\ & \quad -\frac{1}{16}=x\end{align*}

34+23(116)3412418241241724=2(116)+56=18+56=324+2024=1724\begin{align*}\frac{3}{4}+\frac{2}{3} \left(-\frac{1}{16}\right) &= 2\left(-\frac{1}{16}\right)+\frac{5}{6}\\ \frac{3}{4}-\frac{1}{24} &= -\frac{1}{8}+\frac{5}{6}\\ \frac{18}{24}-\frac{1}{24} &= -\frac{3}{24}+\frac{20}{24} \\ \frac{17}{24}&=\frac{17}{24}\end{align*}

#### Example 3

Solve the following equation: 0.6(2x7)=5x5.1\begin{align*}0.6(2x-7)=5x-5.1\end{align*}. Check your answer.

Even though there are decimals in this problem, we can approach it like any other problem. Use the Distributive Property and combine like terms.

0.6(2x+7)=4.3x5.1  1.2x+4.2=4.3x5.14.3x4.2 4.3x4.2 3.1x3.1=9.33.1  x=3\begin{align*}& \ 0.6(2x+7)=4.3x-5.1\\ & \ \ 1.2x+\cancel{4.2}=\cancel{4.3x}-5.1\\ & \underline{-4.3x-\cancel{4.2} \ -\cancel{4.3x}-4.2 \; \;}\\ & \quad \quad \ \frac{-\cancel{3.1}x}{-\cancel{3.1}}=\frac{-9.3}{-3.1}\\ & \qquad \qquad \ \ x=3\end{align*}

0.6(2(3)+7)0.6137.8=4.3(3)5.1=12.95.1=7.8\begin{align*}0.6(2(3)+7) &= 4.3(3)-5.1\\ 0.6 \cdot 13 &= 12.9-5.1 \\ 7.8 &= 7.8\end{align*}

### Review

Solve each equation and check your solution.

1. 6(2x5)=18\begin{align*}-6(2x-5)=18\end{align*}
2. 2(3x+7)=x+19\begin{align*}2-(3x+7)=-x+19\end{align*}
3. 3(x4)=5(x+6)\begin{align*}3(x-4)=5(x+6)\end{align*}
4. x45=23x+815\begin{align*}x-\frac{4}{5}=\frac{2}{3}x+\frac{8}{15}\end{align*}
5. 89x5=x+23\begin{align*}8-9x-5=x+23\end{align*}
6. x12+3x=2(x+18)\begin{align*}x-12+3x=-2(x+18)\end{align*}
7. 52x+14=34x+2\begin{align*}\frac{5}{2}x+\frac{1}{4}=\frac{3}{4}x+2\end{align*}
8. \begin{align*}5 \left(\frac{x}{3}+2\right)=\frac{32}{3}\end{align*}
9. \begin{align*}7(x-20)=x+4\end{align*}
10. \begin{align*}\frac{2}{7}\left(x+\frac{2}{3}\right)=\frac{1}{5}\left(2x-\frac{2}{3}\right)\end{align*}
11. \begin{align*}\frac{1}{6}(x+2)=2\left(\frac{3x}{2}-\frac{5}{4}\right)\end{align*}
12. \begin{align*}-3(-2x+7)-5x=8(x-3)+17\end{align*}

Challenge Solve the equations below. Check your solution.

1. \begin{align*}\frac{3x}{16}+\frac{x}{8}=\frac{3x+1}{4}+\frac{3}{2}\end{align*}
2. \begin{align*}\frac{x-1}{11}=\frac{2}{5} \cdot \frac{x+1}{3}\end{align*}
3. \begin{align*}\frac{3}{x}=\frac{2}{x+1}\end{align*}

To see the Review answers, open this PDF file and look for section 1.9.

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