### Systems of Equations Word Problems Worksheets

In solving word problems, there are actually no definite fastest rules to arrive at the answer. However, there are general helpful steps to follow in solving certain problems. These steps are believed to be just common sense made precise.

Below are helpful steps to follow when solving word problems involving systems of equations in two variables:

1. Understand the problem.
• Know what are the given quantities?
• What is (are) the unknown(s)?
• What are the conditions?
2. Devise a plan. It can be a model/drawing, a table or an equation.
3. Carry out the plan. Solve the equation and look for the unknown quantities.
4. Look back and check if the answers obtained are reasonable.
Examples

Solve each problem.

1. A pharmacist has 30L of a 10% drug solution. How many liters of 5% solution must be added to get a mixture that is 8%?

Solution:

Let be the number of liters of 5% solution and be the number of liters in the mixture that is 8% solution. The total mixture contains 30L, so the first equation is $$x+30=y$$. Multiply each volume by the concentration to get the second equation. The system is:
$\small \displaystyle \left\{\begin{matrix} x+30=y\\ 0.05x+0.1\left ( 30 \right )=0.08y \end{matrix}\right$
Solving the system by using substitution,
$$0.05x+0.1\left(30 \right)=0.08\left(x+30\right)$$
To undo decimal numbers, multiply both sides by 100.
\small \begin{align*} 5x+10\left ( 30 \right )&=8\left ( x+30 \right )\\5x+300&=8x+240 &&\textup{Use distributive property}\\5x-8x&=240-300 &&\textup{Combine like terms}\\-3x&=-60 &&\textup{Multiply both sides by -1/3}\\\therefore x&=20\end{align*}
Therefore, 20 Liters of 5% must be added to 30L of 10% to get a mixture that is 8%.

2. A movie star is 4 times as old as his daughter. Seven years ago he was 11 times as old as she was. How old is the movie star?

Solution:

Let $$D$$ be the daughter’s age and $$M$$ be the movie star’s age. The equations are:
\small \begin{align*}M&=4D\\M-7&=11\left ( D-7 \right ) \end{align*}
Solving the system by substitution, we have:
\small \begin{align*} 4D-7&=11\left ( D-7 \right )&&\textup{Substitute M with 4D}\\4D-7&=11D-77 && \textup{Use distributive property}\\4D-11D&=-77+7&& \textup{Combine like terms}\\ -7D&=-70&&\textup{Divide both sides by -1/7}\\ \therefore D&=10 \end{align*}
Therefore, the movie star is 40 years old.

3. Jewel inherited P500,000 pesos and invested it in two certificates of deposit with simple interest. A portion of the amount was invested at 4½ % and the rest at 6%. If her yearly income from these two investments amounted to P25800, how much money did she invest at each rate?

Solution:

Let $$x$$ be the amount invested at $$4\frac{1}{2}\%$$ and $$y$$ be the amount invested at 6%. Recall that simple interest is obtained by multiplying the principal with the rate and time. The system is:
$\small \displaystyle \left\{\begin{matrix} x+y=500000\\0.045x+0.06y=25800 \end{matrix}\right$

Solve $$x+y=5000000$$ for $$x$$ in terms of $$y$$ and replace $$x$$ with the result in $$0.045x+0.06y=25800$$. Then solve for $$y$$. Use this value to solve for $$x$$.
\small \begin{align*} x&=500000-y\\0.045\left ( 500000-y \right )&=25800\\22500-0.045y+0.06y&=25800\\0.015y&=25800-22500\\0.015y&=3300\\y&=220000\\ \therefore x&=280000 \end{align*}
Therefore, Jewel invested P220,000 at 6% and P280,000 at 4 1/2%.

Practice Exercises

Solve each problem.
1. The perimeter of a triangle is 34 inches. The middle side is twice as long as the shortest side. The longest side is 2 inches less than three times the shortest side. Find the lengths of the three sides.
2. A rectangular garden is 25 ft. wide. If its area is 1125 sq. ft., what is the length of the garden?
3. When 200ml acid solution is added to 300ml of water solution, the result is 19% acid solution. However, when 100ml of the 1st solution is added to 400ml of the 2nd solution, the result is 16% acid solution. Find the concentration of the acid solution.
4. A mother is 25 years older than her daughter. In six years, she will be 3 years more than twice her daughter’s age. Find the present age of the mother and her daughter.
5. A cashier has a total of 126 bills in P1000 and P500. The total value of the money is P101,000.00. How many of each type of bill does she have?