Systems Word Problems
A free Algebra I lesson from the “Systems of Linear Equations and Inequalities” unit, with a worked example and practice problems including step-by-step solutions.
Systems word problems describe two relationships at once. Define variables clearly, translate each relationship into an equation, then solve and interpret the pair.
What you'll learn
- Define variables for two quantities
- Write two equations from a context
- Interpret the solution in the original situation
Worked example
Problem. Tickets cost $8 for adults and $5 for students. Ten tickets cost $68. How many adult tickets were sold?
- Let a be adult tickets and s be student tickets.
- a + s = 10 and 8a + 5s = 68.
- Substitute s = 10 - a: 8a + 5(10 - a) = 68, so a = 6.
Answer: 6
Practice problems
1. Adult tickets cost $8, student tickets cost $5, and 10 tickets cost $68. How many adult tickets?
Show solution
- Warm-up: First identify exactly what the question is asking: Adult tickets cost $8, student tickets cost $5, and 10 tickets cost $68. How many adult tickets?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Use a + s = 10 and 8a + 5s = 68.
- Solving gives a = 6.
- Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.
Answer: 6
2. With the same ticket problem, how many student tickets?
Show solution
- Warm-up: First identify exactly what the question is asking: With the same ticket problem, how many student tickets?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- There are 10 tickets total.
- 10 - 6 = 4.
- Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.
Answer: 4
3. If n + d = 12 describes coins, what does 12 represent?
Choices: Total number of coins · Total cents · Number of nickels only · Number of dimes only
Show solution
- Warm-up: First identify exactly what the question is asking: If n + d = 12 describes coins, what does 12 represent?
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- The equation adds the counts of two coin types.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Total number of coins
4. A gym sells 7 day passes and monthly passes total. Day passes are $10, monthly passes are $40, and revenue is $160. How many monthly passes?
Show solution
- Core Practice: First identify exactly what the question is asking: A gym sells 7 day passes and monthly passes total. Day passes are $10, monthly passes are $40, and revenue is $160. How many monthly passes?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Let d + m = 7 and 10d + 40m = 160.
- Solving gives m = 3.
- Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.
Answer: 3
5. A class has 24 students. There are 4 more girls than boys. How many boys?
Show solution
- Core Practice: First identify exactly what the question is asking: A class has 24 students. There are 4 more girls than boys. How many boys?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Let g = b + 4 and b + g = 24.
- b + b + 4 = 24, so b = 10.
- Check the result by substituting or estimating: the response should match 10 and make sense in the original problem.
Answer: 10
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