Systems of Equations: Word Problems
🧠 What Are Systems Word Problems?
A system of equations is when you have two or more equations working together. We use them to solve real-world problems where there are two unknown values that are related to each other in different ways. For example, figuring out how many adults and children are in a group based on ticket prices and total sales.
📝 How to Solve Systems Word Problems
- Step 1: Define Your Variables. Figure out what the two unknowns are and assign them letters (like x and y).
- Step 2: Write the Equations. Translate the words in the problem into two separate mathematical equations.
- Step 3: Choose a Solution Method. Decide whether to use substitution (solving for one variable and plugging it in) or elimination (adding or subtracting equations to cancel a variable).
- Step 4: Solve the System. Find the values of both variables.
- Step 5: Check Your Answer. Plug your solution back into the original word problem to make sure it makes sense!
🔍 Worked Examples
Example 1: The Concert Tickets
Problem: Tickets for a concert are $5 for students and $10 for adults. 100 tickets were sold, and the total revenue was $650. How many of each ticket were sold?
Solution:
- Define: Let s = number of student tickets, a = number of adult tickets.
- Equations:
Equation 1 (Total Tickets): s + a = 100
Equation 2 (Total Money): 5s + 10a = 650 - Solve (using substitution):
Solve the first equation for s: s = 100 - a
Substitute into the second: 5(100 - a) + 10a = 650
Simplify: 500 - 5a + 10a = 650 → 500 + 5a = 650
Solve: 5a = 150 → a = 30
Then: s = 100 - 30 = 70 - Answer: 70 student tickets and 30 adult tickets.
⚠️ Common Mistakes to Avoid
- Mixing up variables: Be clear about what x and y stand for. Write it down!
- Incorrect equations: Double-check that your equations accurately represent the problem. Does your money equation use the correct prices?
- Forgetting to find both values: After you find x, don't forget to plug it back in to find y.
- Not checking the solution: Always verify your answer makes sense in the context of the story.
💡 Tips & Tricks
- Look for "total" phrases: Phrases like "the total number is..." often give you your first equation (x + y = total).
- Look for value phrases: Phrases involving money or value give you your second equation ((price1)(x) + (price2)(y) = total value).
- Use elimination when the coefficients of one variable are the same or opposites.
- Use substitution when one variable is already isolated or easily isolated.
🎯 How to Practice
Start with simple problems involving two types of items (tickets, coins, people). Practice writing the equations from the words before you even try to solve them. Create your own word problems for a friend to solve! Use online platforms like Khan Academy for instant feedback on your practice.