Linear system word problems (ACT/SAT). Game on.

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.

CCSS HSA.CED.A.310 questions in the bank
Kickoff

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Warm-upeasy

At a cafe, burgers cost bb dollars each and fries cost ff dollars each. One order of 44 burgers and 33 orders of fries cost $30.\text{\char36}30\textsf{.} Another order of 22 burgers and 55 orders of fries cost $22.\text{\char36}22\textsf{.} What is the price of one order of fries?

The system is 4b+3f=304b + 3f = 30 and 2b+5f=22.2b + 5f = 22\textsf{.} Double the second equation: 4b+10f=44.4b + 10f = 44\textsf{.} Subtract the first: 7f=14,7f = 14\textsf{,} so f=2.f = 2\textsf{.}

Mid-gameeasy

A snack bar sells large drinks for LL dollars each and small drinks for SS dollars each. One purchase of 22 large drinks and 33 small drinks cost $19.\text{\char36}19\textsf{.} Another purchase of 44 large drinks and 11 small drink cost $23.\text{\char36}23\textsf{.} What is the price of one small drink?

The system is 2L+3S=192L + 3S = 19 and 4L+S=23.4L + S = 23\textsf{.} Double the first equation: 4L+6S=38.4L + 6S = 38\textsf{.} Subtract the second: 5S=15,5S = 15\textsf{,} so S=3.S = 3\textsf{.}

Mid-gameeasy

A garden store sells large planters for LL dollars each and small planters for SS dollars each. One customer bought 22 large planters and 33 small planters for $54.\text{\char36}54\textsf{.} Another customer bought 11 large planter and 55 small planters for $55.\text{\char36}55\textsf{.} What is the price of one small planter?

The system is 2L+3S=542L + 3S = 54 and L+5S=55.L + 5S = 55\textsf{.} Double the second equation: 2L+10S=110.2L + 10S = 110\textsf{.} Subtract the first: 7S=56,7S = 56\textsf{,} so S=8.S = 8\textsf{.}

Mid-gamemedium

At a fruit stand, apples cost aa dollars per pound and oranges cost oo dollars per pound. One customer bought 44 pounds of apples and 22 pounds of oranges for $14.\text{\char36}14\textsf{.} Another customer bought 22 pounds of apples and 55 pounds of oranges for $19.\text{\char36}19\textsf{.} What is the price of one pound of oranges?

The system is 4a+2o=144a + 2o = 14 and 2a+5o=19.2a + 5o = 19\textsf{.} Double the second equation: 4a+10o=38.4a + 10o = 38\textsf{.} Subtract the first: 8o=24,8o = 24\textsf{,} so o=3.o = 3\textsf{.}

Mid-gamemedium

A bakery sells muffins for mm dollars each and cookies for cc dollars each. One customer bought 33 muffins and 55 cookies for $22.\text{\char36}22\textsf{.} Another customer bought 22 muffins and 22 cookies for $12.\text{\char36}12\textsf{.} What is the price of one muffin?

The system is 3m+5c=223m + 5c = 22 and 2m+2c=12.2m + 2c = 12\textsf{.} Divide the second equation by 2:2\textsf{:} m+c=6,m + c = 6\textsf{,} so c=6m.c = 6 - m\textsf{.} Substitute into the first: 3m+5(6m)=22,3m + 5(6 - m) = 22\textsf{,} so 3m+305m=22,3m + 30 - 5m = 22\textsf{,} 2m=8,-2m = -8\textsf{,} and m=4.m = 4\textsf{.}

Mid-gamemedium

A camp sells senior passes for ss dollars each and junior passes for jj dollars each. One group bought 22 senior passes and 44 junior passes for $50.\text{\char36}50\textsf{.} Another group bought 33 senior passes and 11 junior pass for $40.\text{\char36}40\textsf{.} What is the price of one junior pass?

The system is 2s+4j=502s + 4j = 50 and 3s+j=40.3s + j = 40\textsf{.} From the second equation, j=403s.j = 40 - 3s\textsf{.} Substitute into the first: 2s+4(403s)=50,2s + 4(40 - 3s) = 50\textsf{,} so 2s+16012s=50,2s + 160 - 12s = 50\textsf{,} 10s=110,-10s = -110\textsf{,} s=11,s = 11\textsf{,} and j=7.j = 7\textsf{.}

Mid-gamemedium

A print shop charges cc dollars per color copy and bb dollars per black-and-white copy. One order of 55 color copies and 22 black-and-white copies cost $12.\text{\char36}12\textsf{.} Another order of 22 color copies and 66 black-and-white copies cost $10.\text{\char36}10\textsf{.} What is the price of one black-and-white copy?

The system is 5c+2b=125c + 2b = 12 and 2c+6b=10.2c + 6b = 10\textsf{.} Multiply the first equation by 3:3\textsf{:} 15c+6b=36.15c + 6b = 36\textsf{.} Subtract the second: 13c=26,13c = 26\textsf{,} so c=2.c = 2\textsf{.} Then 2(2)+6b=10,2(2) + 6b = 10\textsf{,} 6b=6,6b = 6\textsf{,} and b=1.b = 1\textsf{.}

Buzzer beatermedium

A clothing shop sells long-sleeve shirts for ll dollars each and short-sleeve shirts for ss dollars each. One purchase of 22 long-sleeve shirts and 33 short-sleeve shirts cost $49.\text{\char36}49\textsf{.} Another purchase of 44 long-sleeve shirts and 11 short-sleeve shirt cost $53.\text{\char36}53\textsf{.} What is the price of one long-sleeve shirt?

The system is 2l+3s=492l + 3s = 49 and 4l+s=53.4l + s = 53\textsf{.} Double the first equation: 4l+6s=98.4l + 6s = 98\textsf{.} Subtract the second: 5s=45,5s = 45\textsf{,} so s=9.s = 9\textsf{.} Then 4l+9=53,4l + 9 = 53\textsf{,} 4l=44,4l = 44\textsf{,} and l=11.l = 11\textsf{.}

Overtime

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