Lesson 4: Problem Solving with Systems

New Concept

Learn to translate complex word problems into solvable systems of equations. We'll model real-world scenarios involving interest, mixtures, and motion by setting up and solving two equations with two variables.

What’s next

Next, we'll walk through examples for interest, mixture, and motion problems. Then you'll solve them yourself in interactive practice cards.

Property

To solve problems involving two investments, use the formula for simple interest, $I = Prt$, where $I$ is the interest, $P$ is the principal, $r$ is the annual interest rate, and $t$ is the time in years. A system of equations is created where one equation represents the sum of the principals, and the second equation represents the sum of the interests earned from each investment.

Total Principal: $P_1 + P_2 = \text{Total P}$
Total Interest: $I_1 + I_2 = \text{Total I}$

Examples

• Maria invests a total of 12,000 dollars in two separate accounts. One account pays 5% annual interest, and the other pays 7%. Her total interest for the year is 700 dollars. If $x$ is the amount at 5% and $y$ is the amount at 7%, the system is $x + y = 12000$ and $0.05x + 0.07y = 700$.

Property

The formula for mixture problems is $P = rW$, where $P$ is the part (amount of pure ingredient), $r$ is the rate (percent concentration), and $W$ is the whole (total amount of the solution). A system is formed by one equation for the total amount of the mixture and a second for the total amount of the key ingredient. Note: We cannot add percents unless they are percents of the same whole amount.

Total Amount: $W_1 + W_2 = W_{\text{total}}$
Amount of Ingredient: $r_1W_1 + r_2W_2 = r_{\text{total}}W_{\text{total}}$

Examples

• A chemist mixes a 15% saline solution with a 40% saline solution to get 100 liters of a 25% solution. Let $x$ be the amount of the 15% solution and $y$ be the amount of the 40% solution. The system is $x + y = 100$ and $0.15x + 0.40y = 0.25(100)$.

Property

Motion problems use the formula $D = RT$, where $D$ is distance, $R$ is rate (speed), and $T$ is time. A system of equations arises when there are two objects in motion or two parts of a journey. The system is built by applying the $D = RT$ formula to each moving object or travel segment.

Examples

• A boat travels 48 miles upstream in 4 hours and makes the return trip downstream in 3 hours. Let $b$ be the boat's speed in still water and $c$ be the current's speed. The system is $4(b - c) = 48$ and $3(b + c) = 48$.

• Two trains leave from the same station at the same time, traveling in opposite directions. One train is 20 mph faster than the other. After 3 hours, they are 420 miles apart. Let $r_1$ and $r_2$ be their speeds. The system is $r_1 = r_2 + 20$ and $3r_1 + 3r_2 = 420$.