Many stories have two unknown amounts. With substitution and what you know about linear models, you can find both.
Why it works
A story with two unknowns usually gives two facts, such as a total count and a total cost. Each fact becomes an equation. The answer must fit both facts at once, which is exactly what the solution of a system does.
Key idea
(1) Name the two unknowns, with units. (2) Write one equation for each fact. (3) Solve the system. (4) Check the answer in the story. (5) Answer in words.
Worked examples
Worked example: Tickets
A school play sold 120 tickets for \$750. Adult tickets cost \$8 and child tickets cost \$5. How many of each were sold?
- Let $a$ = adult tickets and $c$ = child tickets.
- Count: $a + c = 120$. Money: $8a + 5c = 750$.
- From the first, $c = 120 - a$. Substitute: $8a + 5(120 - a) = 750$, so $3a + 600 = 750$ and $a = 50$.
- Then $c = 70$. Check: $8 \cdot 50 + 5 \cdot 70 = 400 + 350 = 750$.
Answer: 50 adult tickets and 70 child tickets
Worked example: When do two plans cost the same?
Gym A charges \$40 to join plus \$15 a month. Gym B charges \$20 to join plus \$20 a month. When do they cost the same?
- Let $x$ = months and $y$ = total cost. A: $y = 15x + 40$. B: $y = 20x + 20$.
- Set them equal: $15x + 40 = 20x + 20$, so $20 = 5x$ and $x = 4$.
- Cost: $15 \cdot 4 + 40 = 100$. After that, A is cheaper, because it adds less each month.
Answer: after 4 months, when both cost \$100
Worked example: Sum and difference
Two numbers add to 31, and one is 7 more than the other. Find them.
- Let $x$ be the larger and $y$ the smaller: $x + y = 31$ and $x = y + 7$.
- Substitute: $(y + 7) + y = 31$, so $2y = 24$ and $y = 12$.
- $x = 19$. Check: $19 + 12 = 31$ and $19 - 12 = 7$.
Answer: 19 and 12
Common mistake
Stopping after finding one unknown, or skipping the check. The question asks for both amounts. Put them back into the story: do they give the right count and the right total?