164 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
The fact that two expressions are equal means that they are equal for any values of the variable. So you can check by taking a few values for the variable and making sure that those do make the two sides equal. Usually we don’t use the values 0 or 1 or 2 as the value for the variable and it is best to avoid numbers that already appear in the problem.
Example 1
Simplify [latex]10(4x+5)[/latex].
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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].
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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].
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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].
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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] |
Caution: Checking your work on problems like these is not completely satisfactory, because students who make a mistake in the original solution often make a corresponding mistake in the checking process. So the fact that it checks does not provide complete confidence that the solution is correct. But it’s still a good idea to check because you’ll catch most of your mistakes.