Section 5.4: Applications & Word Problems

Many real-life problems can be modeled using systems of linear equations. The key is to define variables clearly and translate the situation into equations.

Example 1

Two numbers have a sum of 12. One number is 2 more than the other. Find the numbers.

Step 1: Let \( x \) = first number, \( y \) = second number.

Step 2: Translate into equations: \( x + y = 12 \) and \( x = y + 2 \).

Step 3: Substitute: \( (y+2) + y = 12 \) → \( 2y + 2 = 12 \) → \( 2y = 10 \) → \( y = 5 \)

Step 4: \( x = y + 2 = 7 \)

Solution: Numbers are 7 and 5.

Example 2

A small theater sells 50 tickets. Adult tickets cost $8 and child tickets cost $5. Total revenue is $350. How many adult and child tickets were sold?

Step 1: Let \( a \) = adult tickets, \( c \) = child tickets.

Step 2: Equations: \( a + c = 50 \), \( 8a + 5c = 350 \)

Step 3: Solve first for \( c = 50 - a \)

Step 4: Substitute: \( 8a + 5(50 - a) = 350 \) → \( 8a + 250 - 5a = 350 \) → \( 3a = 100 \) → \( a = 33\frac{1}{3} \)

Step 5: Round to nearest whole number if appropriate: 33 adult tickets, 17 child tickets.

Practice Problems

  1. The sum of two numbers is 20. One is 4 more than the other. Find the numbers.
  2. A store sells pencils for $0.50 each and erasers for $0.75 each. If 40 items are sold for $25, how many pencils and erasers were sold?
  3. A father is 30 years older than his son. Their combined age is 70. Find their ages.
  4. Two trains start from opposite stations 200 km apart. One travels 60 km/h, the other 40 km/h. How long until they meet?
  5. A movie theater sells 120 tickets. Adult tickets cost $10, student tickets $6. Total revenue is $1000. Find number of each ticket sold.
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