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Word Problems & Real-World Applications of Linear Equations

1️⃣ Understanding Word Problems in Algebra

In everyday life, many situations can be modeled using linear equations. Word problems help us apply algebra to real-world challenges in areas such as business, finance, motion, mixtures, and geometry.

To solve word problems effectively, follow these steps:

  1. Read the problem carefully to understand the situation.
  2. Define the variables (let \( x \) or \( y \) represent unknown values).
  3. Set up equations based on the given information.
  4. Solve the system of equations using substitution, elimination, or another method.
  5. Interpret the answer in the context of the problem.
  6. Check your answer to ensure it makes sense.

Example: A school sells adult and child tickets for a play. An adult ticket costs \$10 and a child ticket costs \$5. If the school collects \$300 from selling 40 tickets, how many of each ticket were sold?

  • Define variables: Let \( x \) = number of adult tickets, \( y \) = number of child tickets.
  • Write equations:
    • Total tickets: \( x + y = 40 \)
    • Total revenue: \( 10x + 5y = 300 \)
  • Solve: \( x = 20 \), \( y = 20 \).
  • Final Answer: 20 adult tickets and 20 child tickets.

2️⃣ Motion Problems (Distance, Speed, and Time)

Motion problems involve distance, speed, and time and follow the formula:

\[ \text{Distance} = \text{Speed} \times \text{Time} \]

Example: A train travels 120 km in the same time a car travels 180 km. If the car is 20 km/h faster than the train, find their speeds.

  • Let \( x \) = train’s speed; then the car’s speed = \( x + 20 \).
  • Since time is equal: \[ \frac{120}{x} = \frac{180}{x+20} \]
  • Solve for \( x \): \( x = 60 \) km/h. Therefore, train: 60 km/h, car: 80 km/h.

3️⃣ Mixture Problems

Mixture problems involve combining different solutions or substances.

Example: A shop sells two types of coffee: one costing \$5 per kg and the other costing \$8 per kg. How many kg of each should be mixed to get 20 kg of coffee that costs \$6.50 per kg?

  • Let \( x \) = kg of \$5 coffee, \( y \) = kg of \$8 coffee.
  • Total weight: \( x + y = 20 \).
  • Total cost: \( 5x + 8y = 6.5 \times 20 = 130 \).
  • Solve the system to find \( x = 10 \) and \( y = 10 \).

4️⃣ Geometry Word Problems

Linear equations are used in geometry problems involving perimeter, area, and angles.

Example: The perimeter of a rectangle is 40 cm. The length is 6 cm more than twice the width. Find the dimensions.

  • Let \( w \) = width and \( l = 2w + 6 \).
  • Perimeter equation: \( 2l + 2w = 40 \). Substitute to get \( 2(2w+6) + 2w = 40 \).
  • Solve: \( 4w + 12 + 2w = 40 \) → \( 6w = 28 \) → \( w = 7 \); then \( l = 2(7) + 6 = 20 \).

Practice Questions – Basic

  1. Solve using graphing: \( y = 2x + 3 \) and \( y = -x + 5 \).
  2. Solve using substitution: \( x + y = 10 \) and \( x = 3y - 2 \).
  3. Solve using elimination: \( 4x + 3y = 20 \) and \( 2x - 3y = 4 \).
  4. Find the solution for: \( 5x - y = 12 \) and \( x + 2y = 8 \).
  5. Determine whether the system \( 3x + 6y = 9 \) and \( x + 2y = 3 \) has one, no, or infinitely many solutions.
  6. A farmer has chickens and cows. If there are 50 animals and 140 legs, how many chickens and cows are there?
  7. A factory produces two products: Product A sells for \$5 per unit and Product B sells for \$8 per unit. If 120 units sold yield \$760, how many of each were sold?
  8. Solve for \( x \) and \( y \): \( 2x + 5y = 19 \) and \( 3x - y = 4 \).
  9. Find two numbers that sum to 25 and whose difference is 5.
  10. A two-digit number’s digits add up to 7. When reversed, the number decreases by 9. Find the original number.

Challenging Problems – Push Your Limits!

  1. A car and a motorcycle start from the same point and travel in opposite directions. The car’s speed is 20 km/h faster than the motorcycle’s. After 3 hours, they are 300 km apart. Find their speeds.
  2. A mixture of two alloys weighs 100 kg. One alloy is 40% iron and the other is 60% iron. The final mixture is 50% iron. How much of each alloy should be used?
  3. A student scores twice as many points in math as in science. Together, they score 90 points. Find the math and science scores.
  4. The sum of the squares of two numbers is 50, and their sum is 10. Find the numbers.
  5. A triangle has angles in the ratio 2:3:5. Find the three angles.

6️⃣ Summary

  • ✅ Word problems help apply algebra to real-world scenarios.
  • ✅ Setting up equations correctly is key to solving them.
  • ✅ Understanding motion, mixtures, and geometry enhances problem-solving skills.

7️⃣ Fantastic Job!

Fantastic work! Ready for the next level? Keep going!

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