Math 1 · Unit 6 · Lesson 6 · part 2 of 4
Two facts, two equations
Many stories give you two facts about two unknowns.
Two facts, two equations
Many stories give you two facts about two unknowns. Each fact is an equation, so the story is a system, and the solution is the pair of numbers that makes both facts true. The most common shape is a total count and a total value.
Tickets: count and money
A show sold 40 tickets. Adult tickets cost $8, kid tickets $5, and the total was $260. How many of each?
Let a be adult tickets and k be kid tickets. Count: a + k = 40. Money: 8a + 5k = 260.
Substitution fits (a lonely a): a = 40 − k. Then 8(40 − k) + 5k = 260, so 320 − 8k + 5k = 260, so 320 − 3k = 260, so −3k = −60, so k = 20. Then a = 20.
20 adult and 20 kid tickets. Check in the story: 20 + 20 = 40 tickets; 160 + 100 = 260 dollars.
Coins: count and cents
A jar has 25 coins, all dimes and quarters, worth $4.00. How many of each?
Count: d + q = 25. Cents: 10d + 25q = 400.
Elimination fits (multiply the count by −10): −10d − 10q = −250. Add: 15q = 150, so q = 10. Then d = 15.
15 dimes and 10 quarters. Check: 25 coins; 150 + 250 = 400 cents.
- 1Name the two unknownsOne letter each, and say what they count.
- 2Write two equationsOne for each fact: usually a count and a value.
- 3Pick the method that fitsSubstitution if a letter is alone or easy to free; elimination if the coefficients line up; graph to see.
- 4Solve, then read it back into the storySay the answer in the story's words, with units.
- 5Check in the storyDo the two numbers give the right count and the right value?