Algebra Help Samples

SAMPLE ALGEBRA PROBLEMS:
Question 1: Four times a number, decreased by 25, is 7 more than 3 times 100. Find the number.
Solution: Let \(x\) be the number we are looking for, then
\[ 4x25=7+3\times 100 \]
\[ \Leftrightarrow 4x=25+7+300=332 \]
\[\Leftrightarrow x=\frac{332}{4}=83 \]
which means that the number is 83.
Question 2: How much water must be evaporated from 6 qt of a 50% antifreeze solution to produce a 75% solution? (For full credit on word problems you must solve an appropriate equation.)
Solution: Let x be the amount (in qt) that needs to be evaporized. Then, the following needs to be solved:
\[\frac{0.5\left( 6 \right)}{6x}=0.75\,\,\,\Rightarrow \,\,\,\frac{3}{6x}=0.75\,\,\,\,\Rightarrow \,\,\,3=0.75\left( 6x \right)\,\,\]
\[\Rightarrow \,\,\,\,3=4.50.75x\,\,\,\,\Rightarrow \,\,\,0.75x=1.5\,\,\,\Rightarrow \,\,x=2\text{ qt}\]
Question 3: Given the system of equations below, solve using any appropriate method:
3x + 6y = 21
5x + y = 20
Solution: Multiplying the second equation by 6:
3x + 6y = 21
30x  6y = 120
and now adding:
33x = 99
which means that x = 3 and then y = 20 + 5x = 20 – 5(3) = 5.
The solution is therefore x = 3 and y = 5.
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