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Algebra 101 - Unit 4

Lesson 23: Applications of Systems (Word Problems)

Word problems involving systems of equations allow us to model real-life situations mathematically. We can solve these problems using graphing, substitution, or elimination.

Steps to Solve Word Problems

  1. Read the problem carefully and identify what you are asked to find.
  2. Define variables for unknown quantities.
  3. Write a system of equations based on the problem.
  4. Solve the system using an appropriate method (graphing, substitution, elimination).
  5. Check the solution in the context of the problem.

Example 1

Problem: A movie theater sells 50 tickets for a play. Adult tickets cost $10, and child tickets cost $6. The total revenue is $380. How many adult and child tickets were sold?

Solution:

  1. Let x = number of adult tickets, y = number of child tickets.
  2. Equation 1: x + y = 50 (total tickets)
  3. Equation 2: 10x + 6y = 380 (total revenue)
  4. Use substitution or elimination. Solve Equation 1 for y: y = 50 - x
  5. Substitute into Equation 2: 10x + 6(50 - x) = 380 → 10x + 300 - 6x = 380 → 4x = 80 → x = 20
  6. Find y: y = 50 - 20 = 30
  7. Answer: 20 adult tickets, 30 child tickets.

Example 2

Problem: A store sells pens for $2 each and notebooks for $5 each. On one day, 40 items were sold for $110. How many pens and notebooks were sold?

Solution:

  1. Let x = pens, y = notebooks
  2. Equation 1: x + y = 40
  3. Equation 2: 2x + 5y = 110
  4. Solve Equation 1 for x: x = 40 - y
  5. Substitute into Equation 2: 2(40 - y) + 5y = 110 → 80 - 2y + 5y = 110 → 3y = 30 → y = 10
  6. Find x: x = 40 - 10 = 30
  7. Answer: 30 pens, 10 notebooks.

Practice Problems

  1. A farmer has chickens and cows. There are 20 animals and 56 legs. How many chickens and cows?
  2. A taxi company charges $3 per mile plus $2 flat fee. Another company charges $4 per mile with no flat fee. After how many miles will the total fare be the same?
  3. A fruit seller sold 60 fruits consisting of apples and oranges. Apples cost $1 each, oranges $0.50 each. Total revenue $40. How many apples and oranges were sold?
  4. A gym sells adult memberships for $30 and student memberships for $20. They sold 50 memberships for a total of $1,200. How many of each type?

Lesson Summary