Systems of Equations Word Problems

Many real-world scenarios involve multiple unknown values that are related to each other. By translating these situations into a system of two linear equations, we can easily solve for both unknowns. Common types of problems include pricing, mixtures, and distance/rate/time.

3 Steps to Solve

1. Define your variables: Choose two letters to represent the two unknown quantities you need to find.
2. Set up the equations: Read the problem carefully and translate the relationships into two separate mathematical equations.
3. Solve the system: Use either the substitution or elimination method to find the values of your variables.

Example 1: Pricing and Quantity

Problem: A store sold $50$ items for a total of \350$. Shirts cost$$5$each and pants cost$$10$ each. How many of each did the store sell?

Step 1: Define Variables Let $s$ = the number of shirts sold. Let $p$ = the number of pants sold.

Step 2: Set Up Equations Equation 1 (Total items): $s + p = 50$ Equation 2 (Total cost): $5s + 10p = 350$

Step 3: Solve Let's use the substitution method. From the first equation, we can express $s$ in terms of $p$: $s = 50 - p$

Substitute this into the second equation: $5(50 - p) + 10p = 350$ $250 - 5p + 10p = 350$ $250 + 5p = 350$ $5p = 100$ $p = 20$

Now plug $p = 20$ back into the first equation: $s + 20 = 50$ $s = 30$

Answer: The store sold $30$ shirts and $20$ pants.

Example 2: Distance, Rate, and Time

Problem: Two trains travel toward each other from cities $600$ km apart at $80$ km/h and $70$ km/h. When do they meet?

Step 1: Define Variables Let $x$ = the distance the first train travels. Let $y$ = the distance the second train travels.

Step 2: Set Up Equations Equation 1 (Total distance): $x + y = 600$ Equation 2 (Equal time): Since they start at the same time and meet, their travel times are equal. Time is Distance divided by Rate ($t = \frac{d}{r}$): $\frac{x}{80} = \frac{y}{70}$

Step 3: Solve Multiply the second equation by $560$ (the least common multiple of $80$ and $70$) to clear the fractions: $7x = 8y$ $7x - 8y = 0$

Now we have a clean system:

1. $x + y = 600$
2. $7x - 8y = 0$

Multiply the first equation by $8$: $8x + 8y = 4800$

Add this to the second equation (Elimination method): $(8x + 8y) + (7x - 8y) = 4800 + 0$ $15x = 4800$ $x = 320$

The first train travels $320$ km. To find the time, divide its distance by its speed: $t = \frac{320}{80} = 4$

Answer: The trains will meet in $4$ hours.

Quick Tips

• Look for totals: Sentences like "a total of 50 items" or "totaling $350" usually represent one side of an equation.
• Check your work: Always plug your final answers back into the original word problem to ensure they make logical sense!