Section 5.9: Real-Life Applications

Here are some examples of how systems of equations are used in real-life scenarios.

Example 1: Business

A company sells chairs and tables. Each chair brings $25 profit, each table $40. If total profit is $500 from 15 items, find number of chairs and tables.

Let \( x \) = chairs, \( y \) = tables.

Equations: \( x + y = 15 \), \( 25x + 40y = 500 \)

Solving: \( y = 15 - x \), \( 25x + 40(15 - x) = 500 \) → \( 25x + 600 - 40x = 500 \) → \( -15x = -100 \) → \( x = 6 \)

\( y = 15 - 6 = 9 \)

Solution: 6 chairs, 9 tables.

Example 2: Mixture

Mixing solutions: 30% salt solution and 50% salt solution to get 40 liters of 40% solution. Find amounts of each solution.

Let \( x \) = 30% solution, \( y \) = 50% solution

Equations: \( x + y = 40 \), \( 0.3x + 0.5y = 16 \)

Solving: \( y = 40 - x \), \( 0.3x + 0.5(40 - x) = 16 \) → \( 0.3x + 20 - 0.5x = 16 \) → \( -0.2x = -4 \) → \( x = 20 \)

\( y = 40 - 20 = 20 \)

Solution: 20 liters of each solution.

Practice Problems

  1. Two friends buy snacks: $2 for soda, $3 for chips. Total $20 for 8 items. How many of each?
  2. A school sells pens and pencils: $1.5 each pen, $1 each pencil. Total sales $45 from 30 items. Find quantities.
  3. Mix 25% and 60% solutions to make 50 liters of 40% solution. How much of each?
  4. A bakery sells small and large cakes. Small $8, large $12. Total revenue $200 from 20 cakes. Find number of each.
  5. Two numbers sum to 30. One is twice the other. Find the numbers.
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