UNIT 5Systems of equationsLesson 3 of 31 Solving Systems by Graphing2 Solving Systems by Substitution and Elimination3 Systems of Equations Word Problems
Systems of Equations Word Problems
A story with two unknowns needs two equations. Write them, solve them, then answer in the story's words.
What you'll be able to do
- Name the two unknowns with x and y.
- Write two equations from the story.
- Solve the system and check it in the story.
- Ask if the answer makes sense.
The big idea
Two unknowns, two facts. A story with two unknown numbers needs two facts. Each fact becomes one equation.
An example. A fair sold 50 rides. Kids' rides cost $2. Adult rides cost $4. It made $140.
Name the unknowns. Let x be kids' rides. Let y be adult rides. Write this down first.
Count one way. The rides add to 50: x + y = 50.
Count the money. The money adds to 140: 2x + 4y = 140.
Solve. Put y = 50 − x into the money equation. You get x = 30 and y = 20.
Check in the story. 30 + 20 = 50 rides. 60 + 80 = 140 dollars. Both facts work.
Does it make sense? You can't sell half a ticket. A negative count is wrong too. If the answer can't happen, say so.
Words you need
- System
- Two equations about the same two unknowns.
- Unknown
- A number the story doesn't tell you. A letter stands for it.
- Mixture
- Two things mixed together, like two juices.
- Current
- Moving water that pushes a boat. It helps one way and slows the other way.
- Break even
- When the money in equals the money spent.
Kinds of system stories
| Story | First equation | Second equation |
|---|---|---|
| Tickets or coins | The count adds up | The money adds up |
| Mixture | The liters add up | The part that is juice adds up |
| With and against a current | Speed with it: b + c | Speed against it: b − c |
| Two plans | Plan A's cost | Plan B's cost |
Worked examples
Example 1
A fair sold 50 rides. Kids' rides cost $2. Adult rides cost $4. The fair made $140. How many of each?
- Let x be kids' rides and y be adult rides.
- Count: x + y = 50. Money: 2x + 4y = 140.
- Put y = 50 − x in the money equation: 2x + 200 − 4x = 140. So −2x = −60 and x = 30.
- Then y = 50 − 30 = 20. Check: 60 + 80 = 140.
Answer: 30 kids' rides and 20 adult rides
Example 2
Make 10 liters of juice that is 40% orange. Use juices that are 25% and 50% orange. How much of each?
- Let x and y be the liters of each juice.
- Liters: x + y = 10.
- Orange juice: 25x + 50y = 40 × 10 = 400.
- Put y = 10 − x in: 500 − 25x = 400. So x = 4 and y = 6.
Answer: 4 liters of the 25% juice and 6 liters of the 50% juice
Example 3
A plane flies 900 km with the wind in 3 hours. The trip back takes 5 hours. Find the plane's speed and the wind's speed.
- Let x be the plane's speed and y the wind's speed.
- With the wind: x + y = 900 ÷ 3 = 300.
- Against it: x − y = 900 ÷ 5 = 180.
- Add the two: 2x = 480. So x = 240 and y = 60.
Answer: The plane goes 240 km/h. The wind is 60 km/h.
Common mistakes
Putting the money in the count equation. The count equation adds counts. The money equation adds dollars.
Mixing cents and dollars. Use one unit in an equation. $0.25 and $5.15, or 25 and 515.
Reporting the wrong letter. Write what x and y stand for before you solve.
Taking an answer that can't happen. Half a ticket or a negative count means the story's numbers don't work.
Practice: 20 questions
20 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- A school play sold 120 tickets. Adult tickets cost $8. Student tickets cost $5. The play made $810. Write a system. Then find how many of each were sold. Word problem
- Ines buys pens at $2 each and notebooks at $3 each. She buys 9 items for $22. Write a system. Say what x and y stand for. Then solve it. Word problem
- Two supplementary angles add up to 180°. The larger angle is 12° less than 3 times the smaller. Write a system. Find both angles. Word problem
- A farm stand sold 40 cartons of eggs. Small cartons cost $3. Large cartons cost $5. It made $164. Mei writes x + y = 164 and 3x + 5y = 40. Find her mistake. Word problem
- A club pays $180 for a T-shirt press plus $6 for each blank shirt. It sells printed shirts for $15 each. Write a cost equation and a revenue equation. How many shirts must it sell to break even? What is that amount? Word problem
Tier 2 Intermediate One more step to think through
- A jar holds 32 coins. All of them are dimes ($0.10) and quarters ($0.25). They are worth $5.15. How many of each? Word problem
- Juice A is 20% orange juice. Juice B is 50% orange juice. Kim mixes them to make 12 liters that are 30% orange juice. How many liters of each does she use? Word problem
- A boat goes 36 km downstream in 2 hours. The same trip upstream takes 3 hours. Find the boat's speed in still water and the speed of the current. Word problem
- A cafe sold 50 drinks: teas at $3 and lattes at $5. It made $194. Ola writes t + l = 50 and 3t + 5l = 194. She finds t = 28. She says 28 lattes were sold. Find her mistake. Word problem
- A garden's length is 3 m more than twice its width. Its perimeter is 54 m. Find the length and the width. Word problem
Tier 3 Advanced Several steps - find the hidden structure
- A club sold 40 tickets. Members paid $6 and guests paid $9. The club says it made $310. Is that possible? Explain. Word problem
- Plan A costs $20 a month plus $0.10 a text. Plan B costs $12 a month plus $0.18 a text. For how many texts is Plan A cheaper? Word problem
- Ana walks at 4 km/h, then jogs at 8 km/h. She covers 20 km in 3 hours. How long does she jog? Word problem
- A pet shop has 30 birds and cats. Birds have 2 legs and cats have 4. Together they have 96 legs. Jon writes b + c = 30 and 2b + 2c = 60. He says there are infinitely many answers. Find his mistake. Word problem
- A can holds 10 liters of paint that is 20% blue. Add x liters of paint that is 80% blue. The new total is y liters, and it is 50% blue. Find x and y. Word problem
Tier 4 Challenge Stretch problems, some above grade level
- In 4 years, Lina will be twice as old as Omar. Four years ago, she was 4 times as old as he was. How old are they now? Word problem
- Pencils cost $3 and pens cost $5. Ama spends exactly $31 on pencils and pens. List every way she could do it. Word problem
- Plan A costs $20 plus $10 a month. Plan B costs $50 plus $5 a month. Plan C costs $80 plus $2 a month. For which whole numbers of months is Plan B the cheapest? Word problem
- A two-digit number with tens digit t and ones digit u is 10t + u. A number's digits add up to 11. Swapping the digits makes a number that is 27 more. Find the number. Word problem
- A club spends at most $120 on pizzas at $12 and salads at $8. At most means less than or equal to. It needs at least 10 items. At least means greater than or equal to. A plan (p, s) is p pizzas and s salads. Which plans work: (6, 4), (9, 2), (4, 5) or (2, 12)? Word problem Above grade level
Answers
Tier 1 Foundations
- x + y = 120 and 8x + 5y = 810. 70 adult and 50 student tickets.. (70, 50) makes both equations true: check it in x + y = 120 and 8x + 5y = 810.
- x + y = 9 and 2x + 3y = 22, with x pens and y notebooks. 5 pens and 4 notebooks.. (5, 4) makes both equations true: check it in x + y = 9 and 2x + 3y = 22.
- x + y = 180 and y = 3x − 12. The angles are 48° and 132°.. (48, 132) makes both equations true: check it in x + y = 180 and y = 3x − 12.
- 18 small and 22 large cartons. Mei swapped the two totals. The 40 cartons go in the count: x + y = 40. The $164 goes in the money: 3x + 5y = 164. (18, 22) makes both equations true: check it in x + y = 40 and 3x + 5y = 164.
- Cost y = 180 + 6x. Revenue y = 15x. 20 shirts, when both are $300.. (20, 300) makes both equations true: check it in y = 180 + 6x and y = 15x.
Tier 2 Intermediate
- 19 dimes and 13 quarters. (19, 13) makes both equations true: check it in x + y = 32 and 0.10x + 0.25y = 5.15.
- 8 liters of Juice A and 4 liters of Juice B. (8, 4) makes both equations true: check it in x + y = 12 and 20x + 50y = 30 · 12.
- The boat goes 15 km/h. The current is 3 km/h.. (15, 3) makes both equations true: check it in x + y = 36/2 and x − y = 36/3.
- 28 teas and 22 lattes. t stands for teas, so 28 is the number of teas. The lattes are 50 − 28 = 22. (28, 22) makes both equations true: check it in x + y = 50 and 3x + 5y = 194.
- The length is 19 m and the width is 8 m.. (19, 8) makes both equations true: check it in x = 2y + 3 and 2x + 2y = 54.
Tier 3 Advanced
- No. The system gives 70/3 guest tickets, which is not a whole number.. (50/3, 70/3) makes both equations true: check it in x + y = 40 and 6x + 9y = 310.
- More than 100 texts. They cost the same, $30, at 100 texts.. (100, 30) makes both equations true: check it in y = 20 + 0.10x and y = 12 + 0.18x.
- She jogs for 2 hours. She walks for 1 hour.. (1, 2) makes both equations true: check it in x + y = 3 and 4x + 8y = 20.
- 12 birds and 18 cats. 2b + 2c = 60 just doubles the count. It says nothing new. The legs give 2b + 4c = 96. (12, 18) makes both equations true: check it in x + y = 30 and 2x + 4y = 96.
- 10 + x = y and 20 × 10 + 80x = 50y. x = 10 and y = 20.. (10, 20) makes both equations true: check it in 10 + x = y and 20 · 10 + 80x = 50y.
Tier 4 Challenge
- Lina is 20 and Omar is 8.. Use the ages they are now: x for Lina and y for Omar. In 4 years: x + 4 = 2(y + 4). Four years ago: x − 4 = 4(y − 4).
- 7 pencils and 2 pens, or 2 pencils and 5 pens. There is only one equation, 3x + 5y = 31. But counts are whole numbers. Only two pairs work.
- 7, 8 and 9 months. Find two crossings. A and B cost the same at 6 months. B and C cost the same at 10 months. B wins only between them.
- 47. So t + u = 11. The swapped number is 10u + t, so 10u + t = 10t + u + 27. That gives u − t = 3. So t = 4 and u = 7.
- (6, 4) and (2, 12). Test each point in both: (6, 4): 104 ≤ 120 is true and 10 ≥ 10 is true. (9, 2): 124 ≤ 120 is false and 11 ≥ 10 is true. (4, 5): 88 ≤ 120 is true and 9 ≥ 10 is false. (2, 12): 120 ≤ 120 is true and 14 ≥ 10 is true.
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