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. In Systems of Linear Equations and Inequalities, students need more than a memorized rule: they need to recognize the structure, select a method, carry out the algebra cleanly, and interpret the answer in a graph, table, equation, or real context. The expanded practice now mixes skill fluency, transfer questions, and cumulative review so the lesson builds durable Algebra I readiness.
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.
- Connect the result back to Systems Word Problems so the method and meaning are both clear.
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.
- Identify the Algebra I structure before choosing a calculation.
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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