Applications and Models with Linear Equations

The six steps used in every example

  1. Read the problem. Identify what to find.

  2. Assign a variable. Define the variable (e.g., let x = length). Use tables/diagrams when helpful.

  3. Write an equation. Convert words into algebra.

  4. Solve. Simplify and isolate the variable.

  5. State the answer. Answer the original question with appropriate units.

  6. Check. Verify conditions with the original problem.

Helpful reminders

  • Convert percentages to decimals before multiplying (e.g., 15% = 0.15).

  • For annual rates, 6 months = 0.5 year; maintain consistent units.

Example 1: Square Dimensions

Problem: Each side of a square increases by 3 cm. New perimeter = 40 cm more than twice the original. Find the original dimensions.

  1. Let x = original side (cm). New side = x + 3.

  2. Equation: 4(x + 3) = 40 + 2x.

  3. Solve: 4x + 12 = 40 + 2x → 2x = 28 → x = 14.

  4. Answer: Original square = 14 cm by 14 cm.

  5. Check: New side = 17 cm; New perimeter = 68 cm. Condition satisfied.

Example 2: Motion

Problem: Maria takes 2 hr; Eduardo takes 2.5 hr, 40 mi farther. Eduardo travels 5 mph faster. Find rates.

  1. Let x = Maria’s rate. x + 5 = Eduardo’s rate.

  2. Equation: 2.5(x + 5) = 2x + 40.

  3. Solve: 2.5x + 12.5 = 2x + 40 → 0.5x = 27.5 → x = 55.

  4. Answer: Maria = 55 mph, Eduardo = 60 mph.

  5. Check: Distances match conditions.

Example 3: Mixtures

Problem: Chemist needs 20% alcohol. Has a 15% and 30% solution. Find how much of the 15% to add to 3 L of the 30%.

  1. Let x = liters of 15%.

  2. Equation: 0.15x + 0.90 = 0.20(x + 3).

  3. Solve: 0.15x + 0.90 = 0.20x + 0.60 → 0.30 = 0.05x → x = 6.

  4. Answer: Mix 6 L of 15% with 3 L of 30%.

  5. Check: Alcohol amounts in solutions match.

Example 4: Investments

Problem: Artist sells for $410,000; invests part at 2.65% for 6 months, rest at 2.91% for 1 year. Total interest = $8,761. Find the investment amounts.

  1. Let x = amount at 2.65%.

  2. Equation: 0.5x(0.0265) + 0.0291(410,000 - x) = 8,761.

  3. Solve: 0.01325x + 11,931 - 0.0291x = 8,761 → x = 200,000.

  4. Answer: $200,000 at 2.65%, $210,000 at 2.91%.

  5. Check: Total interest calculated confirms $8,761.