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Tuesday, July 3, 2012

Solving Equations Using Multiplication and Division

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 Solving Equations Using Multiplication and Division

Suppose you are selling pizza for $1.50 a slice and you can get eight slices out of a single pizza. How much money do you get for a single pizza? It shouldn’t take you long to figure out that you get 8 \times \$1.50 = \$12.00. You solved this problem by multiplying. Here’s how to do the same thing algebraically, using x to stand for the cost in dollars of the whole pizza.
Example 4
Solve \frac{1}{8} \cdot x = 1.5.
Our x is being multiplied by one-eighth. To cancel that out and get x by itself, we have to multiply by the reciprocal, 8. Don’t forget to multiply both sides of the equation.
8 \left ( \frac{1}{8} \cdot x \right ) &= 8(1.5)\\ x &= 12
Example 5
Solve \frac{9x}{5} = 5.
\frac{9x}{5} is equivalent to \frac{9}{5} \cdot x, so to cancel out that \frac{9}{5}, we multiply by the reciprocal, \frac{5}{9}.
\frac{5}{9} \left ( \frac{9x}{5} \right ) &= \frac{5}{9}(5)\\ x &= \frac{25}{9}
Example 6
Solve 0.25x = 5.25.
0.25 is the decimal equivalent of one fourth, so to cancel out the 0.25 factor we would multiply by 4.
4(0.25x) &= 4(5.25)\\ x &= 21
Solving by division is another way to isolate x. Suppose you buy five identical candy bars, and you are charged $3.25. How much did each candy bar cost? You might just divide $3.25 by 5, but let’s see how this problem looks in algebra.
Example 7
Solve 5x = 3.25.
To cancel the 5, we divide both sides by 5.
\frac{5x}{5} &= \frac{3.25}{5}\\ x &= 0.65
Example 8
Solve 7x = \frac{5}{11}.
Divide both sides by 7.
x &= \frac{5}{11.7}\\ x &= \frac{5}{77}
Example 9
Solve 1.375x = 1.2.
Divide by 1.375
x &= \frac{1.2}{1.375}\\ x &= 0.8 \overline{72}
Notice the bar above the final two decimals; it means that those digits recur, or repeat. The full answer is 0.872727272727272...

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