Linear system word problems (ACT/SAT) is a grade 9 math skill aligned to Common Core standard HSA.CED.A.3: represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Below are 8 practice questions with answers and step-by-step explanations, drawn from the 10 linear system word problems (act/sat) problems our math games drill.
At a cafe, burgers cost b dollars each and fries cost f dollars each. One order of 4 burgers and 3 orders of fries cost $30. Another order of 2 burgers and 5 orders of fries cost $22. What is the price of one order of fries?
The system is 4b+3f=30 and 2b+5f=22. Double the second equation: 4b+10f=44. Subtract the first: 7f=14, so f=2.
2
Mid-gameeasy
A snack bar sells large drinks for L dollars each and small drinks for S dollars each. One purchase of 2 large drinks and 3 small drinks cost $19. Another purchase of 4 large drinks and 1 small drink cost $23. What is the price of one small drink?
The system is 2L+3S=19 and 4L+S=23. Double the first equation: 4L+6S=38. Subtract the second: 5S=15, so S=3.
3
Mid-gameeasy
A garden store sells large planters for L dollars each and small planters for S dollars each. One customer bought 2 large planters and 3 small planters for $54. Another customer bought 1 large planter and 5 small planters for $55. What is the price of one small planter?
The system is 2L+3S=54 and L+5S=55. Double the second equation: 2L+10S=110. Subtract the first: 7S=56, so S=8.
4
Mid-gamemedium
At a fruit stand, apples cost a dollars per pound and oranges cost o dollars per pound. One customer bought 4 pounds of apples and 2 pounds of oranges for $14. Another customer bought 2 pounds of apples and 5 pounds of oranges for $19. What is the price of one pound of oranges?
The system is 4a+2o=14 and 2a+5o=19. Double the second equation: 4a+10o=38. Subtract the first: 8o=24, so o=3.
5
Mid-gamemedium
A bakery sells muffins for m dollars each and cookies for c dollars each. One customer bought 3 muffins and 5 cookies for $22. Another customer bought 2 muffins and 2 cookies for $12. What is the price of one muffin?
The system is 3m+5c=22 and 2m+2c=12. Divide the second equation by 2:m+c=6, so c=6−m. Substitute into the first: 3m+5(6−m)=22, so 3m+30−5m=22,−2m=−8, and m=4.
6
Mid-gamemedium
A camp sells senior passes for s dollars each and junior passes for j dollars each. One group bought 2 senior passes and 4 junior passes for $50. Another group bought 3 senior passes and 1 junior pass for $40. What is the price of one junior pass?
The system is 2s+4j=50 and 3s+j=40. From the second equation, j=40−3s. Substitute into the first: 2s+4(40−3s)=50, so 2s+160−12s=50,−10s=−110,s=11, and j=7.
7
Mid-gamemedium
A print shop charges c dollars per color copy and b dollars per black-and-white copy. One order of 5 color copies and 2 black-and-white copies cost $12. Another order of 2 color copies and 6 black-and-white copies cost $10. What is the price of one black-and-white copy?
The system is 5c+2b=12 and 2c+6b=10. Multiply the first equation by 3:15c+6b=36. Subtract the second: 13c=26, so c=2. Then 2(2)+6b=10,6b=6, and b=1.
8
Buzzer beatermedium
A clothing shop sells long-sleeve shirts for l dollars each and short-sleeve shirts for s dollars each. One purchase of 2 long-sleeve shirts and 3 short-sleeve shirts cost $49. Another purchase of 4 long-sleeve shirts and 1 short-sleeve shirt cost $53. What is the price of one long-sleeve shirt?
The system is 2l+3s=49 and 4l+s=53. Double the first equation: 4l+6s=98. Subtract the second: 5s=45, so s=9. Then 4l+9=53,4l=44, and l=11.
Overtime
2 more. Played, not assigned.
The rest of the bank lives inside live head-to-head matches — the placement test finds the right starting difficulty.