# B1.02: Section 1

Use the distributive property and other properties to simplify expressions and check your work.

### Distributive Property:

For any real numbers*a*,

*b*, and

*c*:

*a*(

*b*+

*c*) =

*ab*+

*ac*. For any real numbers

*a*,

*b*, and

*c*:

*a*(

*b*–

*c*) =

*ab*–

*ac*

### Example 1

Simplify [latex]10(4x+5)[/latex].Answer:

Solution: [latex]10(4x+5)=40x+50[/latex] | Partial Check: Use [latex]\begin{align}&\,\,\,\,\,\,10(4x+5)=40x+50\\&10(4\cdot3+5)\,\,?=?\,40\cdot3+50\\&\,10(12+5)\,\,\,?=?\,\,120+50\\&\,\,\,\,\,\,\,10(17)\,\,\,?=?\,\,170\\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,170\,\,\,\,=\,\,\,\,170\\\end{align}[/latex] |

### Example 2

Simplify [latex]-6(x+9)[/latex].Answer:

Solution: [latex]-6(x+9)=-6x-54[/latex] | Partial Check: Use [latex]x=5[/latex] [latex]\begin{align}&\,\,\,\,-6(x+9)=-6x-54\\&-6(5+9)\,\,\,?=?\,\,\,-6*5-54\\&\,\,\,-6(14)\,\,\,?=?\,\,\,-30-54\\&\,\,\,\,\,\,\,\,\,\,\,\,\,-84\,\,=\,\,-84\\\end{align}[/latex] |

### Example 3

Simplify [latex]2(3x-7)[/latex].Answer:

Solution: [latex]\begin{align}&2(3x-7)=2\cdot3x-2\cdot7\\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=6x-14\\\end{align}[/latex] | Partial Check: Use [latex]x=5[/latex] [latex]\begin{align}&2(3\cdot5-7)\,\,\,\,\,?=?\,\,\,\,6\cdot5-14\\&\,\,2(15-7)\,\,\,\,?=?\,\,\,30-14\\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2(8)\,\,\,=\,\,\,16\\\end{align}[/latex] |

### Example 4

Simplify [latex]17-5(4x-12)[/latex].Answer:

Solution: [latex]\begin{align}&17-5(4x-12)=17-20x-(-60)\\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=17-20x+60\\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=77-20x\\\end{align}[/latex] | Partial Check: Use [latex]x=3[/latex] [latex]\begin{align}&17-5(4\cdot3-12)\,\,\,?=?\,\,\,77-20\cdot3\\&\,\,\,17-5(12-12)\,\,\,?=?\,\,77-60\\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,17-0\,\,\,=\,\,\,\,17\\\end{align}[/latex] |

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### CC licensed content, Shared previously

- Mathematics for Modeling.
**Authored by:**Mary Parker and Hunter Ellinger.**License:**CC BY: Attribution.