Sample ALgebra Solved 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

\[ 4x-25=7+3\times 100 \]

\[ \Leftrightarrow 4x=25+7+300=332 \]

\[\Leftrightarrow x=\frac{332}{4}=83 \]

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:

\[\frac{0.5\left( 6 \right)}{6-x}=0.75\,\,\,\Rightarrow \,\,\,\frac{3}{6-x}=0.75\,\,\,\,\Rightarrow \,\,\,3=0.75\left( 6-x \right)\,\,\]

\[\Rightarrow \,\,\,\,3=4.5-0.75x\,\,\,\,\Rightarrow \,\,\,0.75x=1.5\,\,\,\Rightarrow \,\,x=2\text{ qt}\]

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.