➕ Algebra 1 · Systems of Equations

Systems Word Problems practice questions

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Answer key with explanations

1. A zoo sold 80 tickets in all. Child tickets cost 5 dollars each and adult tickets cost 9 dollars each, and the sales totaled 520 dollars. Let c be the number of child tickets and a be the number of adult tickets. Which pair of equations models the story?

   Answer: c + a = 80 and 5c + 9a = 520. “80 tickets in all” is a count, so c + a = 80. The money is price × number for each kind: 5c + 9a = 520 dollars. The other choices swap the totals (mixing tickets with dollars), swap the prices, or use 80 as a dollar total. The solution is 50 child tickets and 30 adult tickets: 50 + 30 = 80 and 5(50) + 9(30) = 250 + 270 = 520.

2. A bakery sold 50 cookies and muffins in one morning. Cookies cost 2 dollars each and muffins cost 3 dollars each, and the morning's sales were 130 dollars. How many muffins were sold?

   Answer: 30 muffins. Let c be the number of cookies and m be the number of muffins. Count: c + m = 50. Dollars: 2c + 3m = 130. Substitute c = 50 − m: 2(50 − m) + 3m = 130, so 100 + m = 130 and m = 30. Then c = 20. Check: 20 + 30 = 50 and 2(20) + 3(30) = 40 + 90 = 130. The choice 20 is the number of cookies, 26 is 130 ÷ (2 + 3), and 43 is about 130 ÷ 3.

3. Plan A rents a scooter for 16 dollars per day. Plan B charges a 24-dollar fee plus 8 dollars per day. A graph of total cost in dollars (vertical axis) against days (horizontal axis) shows the two lines crossing at (3, 48). What does the crossing point mean?

   Answer: After 3 days, both plans cost 48 dollars in total. A point is written (horizontal, vertical), and here the horizontal axis is days and the vertical axis is dollars. So (3, 48) means 3 days and 48 dollars. Check both formulas: Plan A costs 16(3) = 48 dollars and Plan B costs 24 + 8(3) = 48 dollars. The other choices swap the coordinates or misread what the crossing shows.

4. Ivy earns 12 dollars per hour babysitting and 9 dollars per hour walking dogs. Last week she worked 15 hours and earned 156 dollars. How many hours did she spend babysitting?

   Answer: 7 hours. Let b be the hours babysitting and w be the hours walking dogs. Hours: b + w = 15. Dollars: 12b + 9w = 156. Substitute w = 15 − b: 12b + 135 − 9b = 156, so 3b = 21 and b = 7. Then w = 8. Check: 7 + 8 = 15 and 12(7) + 9(8) = 84 + 72 = 156. The choice 8 is the dog-walking hours, 13 is 156 ÷ 12 (as if she only babysat), and 21 adds the two pay rates.

5. Phone plan A has a 25-dollar activation fee and costs 10 dollars per month. Phone plan B has a 5-dollar activation fee and costs 15 dollars per month. After how many months do the two plans have the same total cost?

   Answer: 4 months. Let t be the number of months. Plan A costs 25 + 10t dollars and plan B costs 5 + 15t dollars. Set them equal: 25 + 10t = 5 + 15t. Subtract 10t: 25 = 5 + 5t. Subtract 5: 20 = 5t. Divide by 5: t = 4. Check: 25 + 40 = 65 and 5 + 60 = 65 dollars. The choice 6 adds the fees instead of subtracting, 5 uses only one fee, and 20 forgets to divide by 5.

6. A vending machine holds only quarters and dimes: 30 coins worth 495 cents in all. (A quarter is worth 25 cents and a dime is worth 10 cents.) How many quarters are in the machine?

   Answer: 13 quarters. Let q be the number of quarters and d be the number of dimes. Count: q + d = 30. Cents: 25q + 10d = 495. Substitute d = 30 − q: 25q + 300 − 10q = 495, so 15q = 195 and q = 13. Then d = 17. Check: 13 + 17 = 30 and 25(13) + 10(17) = 325 + 170 = 495. The choice 17 is the number of dimes, and 15 guesses an even split.

7. A boat travels 70 kilometers downstream in 5 hours, then returns the same 70 kilometers upstream in 7 hours. Assume the boat's speed through the water and the speed of the current both stay steady. What is the speed of the current?

   Answer: 2 kilometers per hour. Let b be the boat's speed through the water and c be the current's speed, both in kilometers per hour. Downstream the current helps, so the speed over the ground is b + c = 70 ÷ 5 = 14. Upstream it hurts, so b − c = 70 ÷ 7 = 10. Add the equations: 2b = 24, so b = 12. Then c = 14 − 12 = 2. Check: (12 + 2)(5) = 70 and (12 − 2)(7) = 70. The choice 12 is the boat's speed, 4 forgets to halve 14 − 10, and 14 is the downstream speed.

8. A food truck says it sold 14 items in one day: wraps at 5 dollars each and smoothies at 3 dollars each, for 51 dollars in all. Which statement is correct?

   Answer: The equations give 4.5 wraps and 9.5 smoothies, which is not possible because items are sold whole. Let w be the number of wraps and s be the number of smoothies. Count: w + s = 14. Dollars: 5w + 3s = 51. Substitute s = 14 − w: 5w + 42 − 3w = 51, so 2w = 9 and w = 4.5. Then s = 9.5. Check: 4.5 + 9.5 = 14 and 22.5 + 28.5 = 51. So the equations do have a solution, but half an item cannot be sold. Rounding does not help: 4 wraps and 10 smoothies is 50 dollars, and 5 wraps and 9 smoothies is 52 dollars. Another way to see it: with whole items, 5w + 3(14 − w) = 42 + 2w is always even, but 51 is odd.

9. Studio X charges a 30-dollar fee plus 12 dollars per class. Studio Y charges an 18-dollar fee plus 12 dollars per class. After how many classes do the two studios cost the same in total?

   Answer: Never: no number of classes makes the totals equal. Let n be the number of classes. Studio X costs 30 + 12n dollars and Studio Y costs 18 + 12n dollars. Set them equal: 30 + 12n = 18 + 12n. Subtract 12n: 30 = 18, which is false. The n-terms vanished and what is left is false, so there is no solution. In the story, the rates are equal, so Studio X costs 12 dollars more for every number of classes. The choice “after 12 classes” treats the fee difference 30 − 18 = 12 as an answer.