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# Distributive Property for Multi-Step Equations

## a(x + b) = c

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Distributive Property for Multi-Step Equations

Suppose you were a contestant on a game show where they gave you the answer and you had to think of the corresponding question. If the answer were, "M(N+K)=MN+MK\begin{align*}M(N+K)= MN+MK\end{align*} or M(NK)=MNMK\begin{align*}M(N-K)= MN-MK\end{align*}," what would you give for the question? How about, "What is the Distributive Property?" In this Concept, you will learn all about the Distributive Property and how to use it to solve equations with multiple steps.

### Guidance

Solving Multi-Step Equations by Using the Distributive Property

When faced with an equation such as 2(5x+9)=78\begin{align*}2(5x+9)=78\end{align*}, the first step is to remove the parentheses. There are two options to remove the parentheses. You can apply the Distributive Property or you can apply the Multiplication Property of Equality. This Concept will show you how to use the Distributive Property to solve multi-step equations.

#### Example A

Solve for x\begin{align*}x\end{align*}: 2(5x+9)=78.\begin{align*}2(5x+9)=78.\end{align*}

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

Apply the Addition Property of Equality: 10x+1818=7818.\begin{align*}10x+18-18=78-18.\end{align*}

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

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

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

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

#### Example B

Solve for n\begin{align*}n\end{align*} when 2(n+9)=6n.\begin{align*}2(n+9)=6n.\end{align*}

Solution:

2(n+9)=5n2n+29=5n2n+18=5n2n+2n+18=2n+5n18=3n1318=133n6=n\begin{align*}&2(n+9)=5n\\ &2\cdot n+2\cdot 9=5n\\ &2n+18=5n\\ &-2n+2n+18=-2n + 5n\\ &18=3n\\ &\frac{1}{3}\cdot18=\frac{1}{3}\cdot 3n\\ &6=n \end{align*}

2(6+9)=5(6)2(15)=3030=30\begin{align*} &2(6+9)=5(6)\\ &2(15)=30\\ &30=30\\ \end{align*}

#### Example C

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

Solution:

3(d+15)18d=03d+31518d=03d+4518d=015d+45=015d+4545=04515d=4511515d=11545d=3\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*}

3(3+15)18(3)=03(3+15)18(3)=03(18)18(3)=05454=00=0\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*}

### Vocabulary

Distributive Property: For any real numbers M, N,\begin{align*}M, \ N,\end{align*} and K\begin{align*}K\end{align*}:

M(N+K)=MN+MKM(NK)=MNMK\begin{align*}&M(N+K)= MN+MK\\ &M(N-K)= MN-MK\end{align*}

The Addition Property of Equality: For all real numbers a,b,\begin{align*}a, b,\end{align*} and c\begin{align*}c\end{align*}:

If a=b\begin{align*}a = b\end{align*}, then a+c=b+c\begin{align*}a + c = b + c\end{align*}.

The Multiplication Property of Equality: For all real numbers a,b\begin{align*}a, b\end{align*}, and c\begin{align*}c\end{align*}:

If a=b\begin{align*}a = b\end{align*}, then a(c)=b(c).\begin{align*}a(c)= b(c).\end{align*}

### Guided Practice

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

Solution:

Step 1: Apply the Distributive Property.

3(2x+5)+2x=732x+35+2x=76x+15+2x=7\begin{align*}&3(2x+5)+2x=7\\ &3\cdot 2x+3\cdot 5+2x=7\\ &6x+15+2x=7\\ \end{align*}

Step 2: Combine like terms.

6x+15+2x=78x+15=7\begin{align*}&6x+15+2x=7\\ &8x+15=7\\ \end{align*}

Step 3: Isolate the variable and its coefficient by using the Addition Property.

8x+15=78x+1515=7158x=8\begin{align*} &8x+15=7\\ &8x+15-15=7-15\\ &8x=-8\\ \end{align*}

Step 4: Isolate the variable by applying the Multiplication Property.

8x=8188x=818188x=818x=1\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 x=1\begin{align*}x=-1\end{align*} into 3(2x+5)=7.\begin{align*}3(2x+5)=7.\end{align*}

3(2(1)x+5)+2(1)=3(2+5)2=3(3)2=92=7.\begin{align*}3(2(-1)x+5)+2(-1)=3(-2+5)-2=3(3)-2=9-2=7.\end{align*}

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

### Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Multi-Step Equations (15:01)

Note that in the next video there IS an error.  At one point the teacher says that -135/6 is in simplest form.  There is a common factor of 3 in both numerator and denominator.  -135/6 can be simplified by that factor of 3 to 45/2, or 22.5.

In 1 – 22, solve the equation.

1. 3(x1)2(x+3)=0\begin{align*}3(x - 1) - 2(x + 3) = 0\end{align*}
2. 7(w+20)w=5\begin{align*}7(w + 20) - w = 5\end{align*}
3. 9(x2)=3x+3\begin{align*}9(x - 2) = 3x + 3\end{align*}
4. 2(5a13)=27\begin{align*}2 \left (5a - \frac{1}{3} \right ) = \frac{2}{7}\end{align*}
5. 29(i+23)=25\begin{align*}\frac{2}{9} \left (i + \frac{2}{3} \right ) = \frac{2}{5}\end{align*}
6. 4(v+14)=352\begin{align*}4 \left (v + \frac{1}{4} \right ) = \frac{35}{2}\end{align*}
7. 22=2(p+2)\begin{align*}22=2(p+2)\end{align*}
8. (m+4)=5\begin{align*}-(m+4)=-5\end{align*}
9. 48=4(n+4)\begin{align*}48=4(n+4)\end{align*}
10. 65(v35)=625\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*}

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