# Distributive Property for Multi-Step Equations

## a(x + b) = c

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Distributive Property for Multi-Step Equations

How would you solve the equation \begin{align*}2(5x+9)=78\end{align*}

### Solving Multi-Step Equations Using the Distributive Property

When solving equations, it is often necessary for you to remove parentheses. The Distributive Property will help you do that.

Recall that the Distributive Property states:

• \begin{align*}M(N+K)= MN+MK\end{align*}
• \begin{align*}M(N-K)= MN-MK\end{align*}

#### Let's solve the following equations:

1.  \begin{align*}2(n+9)-5n=0\end{align*}

\begin{align*}&2(n+9)-5n=0\\ &2\cdot n+2\cdot 9-5n=0\\ &2n+18-5n=0\\ &-3n+18=0\\ &-3n=-18\\ &\frac{1}{3}\cdot-3n=\frac{1}{3}\cdot -18\\ &n=6 \end{align*}

Checking the solution:

\begin{align*} &2(6+9)-5(6)=0\\ &2(15)-30=0\\ &0=0\\ \end{align*}

1. \begin{align*}3(d+15)-18d=0.\end{align*}

\begin{align*}&3(d+15)-18d=0\\ &3\cdot d+3\cdot 15-18d=0\\ &3d+45-18d=0\\ &-15d+45=0\\ &-15d+45-45=0-45\\ &-15d=-45\\ &-\frac{1}{15}\cdot -15d=-\frac{1}{15}\cdot-45\\ & d=3\\ \end{align*}

Checking the solution:

\begin{align*}&3(3+15)-18(3)=0\\ &3(3+15)-18(3)=0\\ &3(18)-18(3)=0\\ &54-54=0\\ &0=0\\ \end{align*}

### Example

#### Example 1

Earlier, you were asked about how you could solve \begin{align*}2(5x+9)=78\end{align*}.

First, you need to remove the parentheses. You could use the Multiplication Property of Equality or the Distributive Property. Here, it is easier to use the Distributive Property.

Apply the Distributive Property: \begin{align*}10x+18=78.\end{align*}

Apply the Addition Property of Equality: \begin{align*}10x+18-18=78-18.\end{align*}

Simplify: \begin{align*}10x=60.\end{align*}

Apply the Multiplication Property of Equality: \begin{align*}10x \div 10 = 60 \div 10.\end{align*}

The solution is \begin{align*}x=6\end{align*}.

Checking the solution:

Does \begin{align*}10(6) + 18 = 78?\end{align*} Yes, so the answer is correct.

#### Example 2

Solve for \begin{align*}x\end{align*} when \begin{align*}3(2x+5)+2x=7.\end{align*}

Apply the Distributive Property:

\begin{align*}&3(2x+5)+2x=7\\ &3\cdot 2x+3\cdot 5+2x=7\\ &6x+15+2x=7\\ \end{align*}

Combine like terms:

\begin{align*}&6x+15+2x=7\\ &8x+15=7\\ \end{align*}

Isolate the variable and its coefficient by using the Addition Property:

\begin{align*} &8x+15=7\\ &8x+15-15=7-15\\ &8x=-8\\ \end{align*}

Isolate the variable by applying the Multiplication Property:

\begin{align*} &8x=-8\\ &\frac{1}{8}\cdot 8x=-8\cdot \frac{1}{8}\\ &\frac{1}{8}\cdot 8x=-8\cdot \frac{1}{8}\\ & x=-1 \end{align*}

Substitute \begin{align*}x=-1\end{align*} into \begin{align*}3(2x+5)=7.\end{align*}

\begin{align*}3(2(-1)x+5)+2(-1)=3(-2+5)-2=3(3)-2=9-2=7.\end{align*}

Therefore, \begin{align*}x=-1.\end{align*}

### Review

In 1 – 22, solve the equation.

1. \begin{align*}3(x - 1) - 2(x + 3) = 0\end{align*}
2. \begin{align*}7(w + 20) - w = 5\end{align*}
3. \begin{align*}9(x - 2) = 3x + 3\end{align*}
4. \begin{align*}2 \left (5a - \frac{1}{3} \right ) = \frac{2}{7}\end{align*}
5. \begin{align*}\frac{2}{9} \left (i + \frac{2}{3} \right ) = \frac{2}{5}\end{align*}
6. \begin{align*}4 \left (v + \frac{1}{4} \right ) = \frac{35}{2}\end{align*}
7. \begin{align*}22=2(p+2)\end{align*}
8. \begin{align*}-(m+4)=-5\end{align*}
9. \begin{align*}48=4(n+4)\end{align*}
10. \begin{align*}\frac{6}{5} \left (v- \frac{3}{5} \right ) = \frac{6}{25}\end{align*}
11. \begin{align*}-10(b-3)=-100\end{align*}
12. \begin{align*}6v + 6(4v+1)=-6\end{align*}
13. \begin{align*}-46=-4(3s+4)-6\end{align*}
14. \begin{align*}8(1+7m)+6=14\end{align*}
15. \begin{align*}0=-7(6+3k)\end{align*}
16. \begin{align*}35=-7(2-x)\end{align*}
17. \begin{align*}-3(3a+1)-7a=-35\end{align*}
18. \begin{align*}-2 \left (n+ \frac{7}{3} \right )=- \frac{14}{3}\end{align*}
19. \begin{align*}- \frac{59}{60} = \frac{1}{6} \left (- \frac{4}{3} r-5 \right )\end{align*}
20. \begin{align*}\frac{4y+3}{7} = 9\end{align*}
21. \begin{align*}(c+3)-2c-(1-3c)=2\end{align*}
22. \begin{align*}5m-3[7-(1-2m)]=0\end{align*}

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

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### Vocabulary Language: English

TermDefinition
distributive property The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, $a(b + c) = ab + ac$.
factor Factors are the numbers being multiplied to equal a product. To factor means to rewrite a mathematical expression as a product of factors.
Variable A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.