Applications of Systems (Word Problems)
What You’ll Learn
Section titled “What You’ll Learn”In this lesson you’ll learn how to set up and solve systems of linear equations from word problems, using any method (graphing, substitution, or elimination).
The Concept
Section titled “The Concept”Word problems with systems involve two unknowns and two related conditions. Steps:
- Read carefully and identify the two unknowns (assign variables, e.g., let x = number of adults, y = number of children)
- Write two equations based on the relationships (usually one for totals/counts, one for values/rates)
- Solve the system using substitution, elimination, or graphing
- Interpret the solution in context
- Check: does the answer make sense (positive numbers, realistic amounts)?
Common types:
- Comparison/break-even (two plans, when equal)
- Mixture (blending two items to get a target)
- Number/age problems (adults + children = total, values = total)
- Work/rate (two rates, combined work)
Worked Example
Section titled “Worked Example”A movie theater sells adult tickets for 12 dollars and child tickets for 8 dollars. One night, 220 tickets are sold for 2,360 dollars total. How many adult and child tickets were sold?
- Let a = adult tickets, c = child tickets
- Equation 1 (total tickets): a + c = 220
- Equation 2 (total revenue): 12a + 8c = 2360
Solve using elimination: Multiply first by 8: 8a + 8c = 1760
Subtract from second: (12a + 8c) − (8a + 8c) = 2360 − 1760 → 4a = 600 → a = 150
Substitute: 150 + c = 220 → c = 70
Solution: 150 adult tickets, 70 child tickets.
Check: 150 + 70 = 220 tickets; 12(150) + 8(70) = 1800 + 560 = 2360 dollars ✓
Real-World Application
Section titled “Real-World Application”Systems solve comparison and mixture problems:
- Two jobs: Job A: 18 dollars/hour + 100-dollar bonus. Job B: 22 dollars/hour. When equal? 18h + 100 = 22h → 100 = 4h → h = 25 hours
- Coffee blend: 5 dollars/pound + 9 dollars/pound to make 7 dollars/pound blend. Let x = pounds of 5-dollar coffee, y = pounds of 9-dollar coffee. x + y = total pounds, 5x + 9y = 7 × total → solve for ratio
- Budget: Rent + food = 1200 dollars. Rent is twice food. r + f = 1200, r = 2f → 2f + f = 1200 → f = 400, r = 800
These help compare options, find balance points, or mix solutions in budgeting, work, or purchases.