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:
- Read the problem carefully to understand the situation.
- Define the variables (let \( x \) or \( y \) represent unknown values).
- Set up equations based on the given information.
- Solve the system of equations using substitution, elimination, or another method.
- Interpret the answer in the context of the problem.
- 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
- Solve using graphing: \( y = 2x + 3 \) and \( y = -x + 5 \).
- Solve using substitution: \( x + y = 10 \) and \( x = 3y - 2 \).
- Solve using elimination: \( 4x + 3y = 20 \) and \( 2x - 3y = 4 \).
- Find the solution for: \( 5x - y = 12 \) and \( x + 2y = 8 \).
- Determine whether the system \( 3x + 6y = 9 \) and \( x + 2y = 3 \) has one, no, or infinitely many solutions.
- A farmer has chickens and cows. If there are 50 animals and 140 legs, how many chickens and cows are there?
- 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?
- Solve for \( x \) and \( y \): \( 2x + 5y = 19 \) and \( 3x - y = 4 \).
- Find two numbers that sum to 25 and whose difference is 5.
- 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!
- 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.
- 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?
- A student scores twice as many points in math as in science. Together, they score 90 points. Find the math and science scores.
- The sum of the squares of two numbers is 50, and their sum is 10. Find the numbers.
- 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!