Solve Real-World System Problems - 8th Grade Math

Two friends start biking toward each other from towns that are 300 miles apart. One bikes at 60 mph and the other bikes at 40 mph. Let $t$ be the time in hours until they meet, and let $d$ be the distance (in miles) the 60 mph biker travels. Which system correctly models the situation, and what is $t$ ?
Answer: System: $d=60t$ and $d+40t=300$ ; $t=3$ Since the bikers move toward each other, the 60 mph biker's distance is $d=60t$ , and the 40 mph biker covers the rest of the 300 miles, so $d+40t=300$ . Substituting gives $60t+40t=300$ , so $100t=300$ and $t=3$ hours. Choice A is wrong because it subtracts the two bikers' distances, as if one were chasing down the other, instead of adding them, which is what happens when two people move toward each other. Choice C is wrong because it averages the two speeds together instead of adding the distances each biker actually covers. Choice D is wrong because it ignores the 40 mph biker's contribution entirely, treating the whole 300 miles as covered by the 60 mph biker alone.
A rectangle has perimeter 40 cm. Its length is 3 cm more than its width. Let $\ell$ be the length and $w$ be the width. What are the dimensions of the rectangle?
Answer: $w=8.5\text{ cm},\ \ell=11.5\text{ cm}$ This question tests setting up and solving systems of equations from real-world scenarios requiring two equations for two unknowns with context interpretation. The process involves: (1) defining variables such as w for width and ℓ for length in cm, (2) writing the perimeter equation: 2w + 2ℓ = 40 or w + ℓ = 20, (3) writing ℓ = w + 3, (4) solving by substitution, (5) interpreting, and (6) verifying. Solving correctly: (w + 3) + w = 20, 2w + 3 = 20, 2w = 17, w = 8.5, ℓ = 11.5. These satisfy: perimeter 2(8.5 + 11.5) = 40 and difference 3. Common errors include forgetting to double sides or misinterpreting 'more than.' To avoid errors: recall perimeter formula, define variables with units, translate relations accurately, solve, accept decimals if sensible, and verify.
A store sells notebooks and pens. A notebook costs $3 and a pen costs $1. A student buys 18 items total and spends $38. Let $n$ be the number of notebooks and $p$ be the number of pens. How many notebooks did the student buy?
Answer: 10 This question tests setting up and solving systems of equations from real-world scenarios requiring two equations for two unknowns with context interpretation. The process involves: (1) defining variables such as $n$ for notebooks and $p$ for pens, (2) writing $n + p = 18$ for total items, (3) writing $3n + p = 38$ for total cost, (4) solving, (5) interpreting (number of notebooks), and (6) verifying. Solving correctly: subtract first from second: $(3n + p) - (n + p) = 38 - 18$ , $2n = 20$ , $n = 10$ , then $p = 8$ . These satisfy: $10 + 8 = 18$ and $3(10) + 8 = 30 + 8 = 38$ . Common errors include incorrect cost equation or not solving the system fully. To avoid errors: define clearly, translate costs and counts, use elimination or substitution, interpret the asked quantity, and verify.
A rectangle has perimeter 40 cm. Its length is 3 cm more than its width. Let $l$ be the length and $w$ be the width. Solve the system to find the dimensions of the rectangle.
Answer: $w=8.5$ cm and $l=11.5$ cm This problem tests setting up and solving systems of equations from real-world scenarios requiring two equations for two unknowns, along with interpreting the solution in context. The process involves: (1) defining variables with units (let l = length in cm, w = width in cm), (2) writing the perimeter equation (2l + 2w = 40 or l + w = 20), (3) writing the relation (l = w + 3), (4) solving by substitution (w + 3 + w = 20, 2w = 17, w = 8.5, l = 11.5), (5) interpreting (width 8.5 cm, length 11.5 cm), and (6) verifying perimeter and relation. The correct system is l + w = 20 and l = w + 3, yielding w = 8.5 cm and l = 11.5 cm. These equations are correct as they simplify the perimeter and capture the dimension difference. Common errors include forgetting to halve the perimeter or reversing the relation, leading to integer but incorrect dimensions like w=7, l=13. To set up correctly: express perimeter properly, define relations clearly, solve accurately, ensure dimensions make sense, and verify. Avoid errors like using incorrect coefficients or not checking units.
Two friends' ages add to 45. One friend is 5 years older than the other. Let $x$ be the older friend's age and $y$ be the younger friend's age. Write a system of equations and solve for $x$ and $y$ .
Answer: $x=25,\ y=20$ This problem tests setting up and solving systems of equations from real-world scenarios requiring two equations for two unknowns, along with interpreting the solution in context. The process involves: (1) defining variables with units (let x = older friend's age in years, y = younger friend's age in years), (2) writing the equation from the sum constraint (x + y = 45), (3) writing the equation from the difference constraint (x = y + 5), (4) solving the system using substitution (substitute into first: y + 5 + y = 45, 2y = 40, y = 20, x = 25), (5) interpreting the results (older friend is 25 years old, younger is 20), and (6) verifying both conditions. The correct system is x + y = 45 and x = y + 5, yielding x = 25 and y = 20. These equations are correct because they capture the total age and the age difference accurately. Common errors include reversing the difference equation or arithmetic slips leading to pairs like x=20, y=25, which don't make sense contextually. To set up correctly: identify the relationships carefully, define variables with units, form equations directly from statements, solve step-by-step, interpret sensibly (ages positive and logical), and verify. Avoid errors like assuming the wrong variable is older or not checking if the solution fits the real-world context.
A gym sells day passes and monthly passes. A day pass costs $6 and a monthly pass costs $30. On one day, the gym sold 40 total passes and collected $720. Let $d$ be the number of day passes and $m$ be the number of monthly passes. Which system represents the situation and what are $d$ and $m$ ?
Answer: System: $d+m=40$ and $6d+30m=720$ ; $d=20,\ m=20$ This question tests setting up and solving systems of equations from real-world scenarios requiring two equations for two unknowns with context interpretation. The process involves: (1) defining variables such as d for day passes and m for monthly passes, (2) writing d + m = 40 for total passes, (3) writing 6d + 30m = 720 for revenue, (4) solving, (5) interpreting, and (6) verifying. Solving correctly: from d = 40 - m, 6(40 - m) + 30m = 720, 240 - 6m + 30m = 720, 240 + 24m = 720, 24m = 480, m = 20, d = 20. These fit both equations. Common errors include swapping costs or totals, like using d + m = 720. To avoid errors: match constraints to equations carefully, solve step-by-step, ensure non-negative integers, and verify.
The sum of two siblings' ages is 45 years, and the older sibling is 5 years older than the younger sibling. Let $o$ be the older sibling's age and $y$ be the younger sibling's age. What are their ages?
Answer: $o=25,\ y=20$ This question tests setting up and solving systems of equations from real-world scenarios requiring two equations for two unknowns with context interpretation. The process involves: (1) defining variables such as o for older sibling's age and y for younger's age in years, (2) writing the equation for sum: o + y = 45, (3) writing the equation for difference: o = y + 5, (4) solving by substitution, (5) interpreting the results, and (6) verifying. Solving correctly: substitute o = y + 5 into the first to get (y + 5) + y = 45, 2y + 5 = 45, 2y = 40, y = 20, o = 25. These satisfy both: 25 + 20 = 45 and 25 - 20 = 5. Common errors include reversing older and younger or misstating the difference, leading to pairs like o=20, y=25. To avoid errors: identify relationships clearly, define variables precisely, solve systematically, check for sensibility (ages positive), and verify both conditions.
A juice mix is made using apple juice and grape juice. The total amount of mix is 10 liters. Apple juice costs $3 per liter and grape juice costs $5 per liter. The total cost is $38. Let $a$ be liters of apple juice and $g$ be liters of grape juice. How many liters of each were used?
Answer: $a=6,\ g=4$ This problem tests setting up and solving systems from real-world scenarios requiring two equations for two unknowns with context interpretation. The process involves: (1) defining variables with units (let a = liters of apple juice, g = liters of grape juice), (2) writing total volume equation (a + g = 10), (3) writing total cost equation (3a + 5g = 38), (4) solving (a = 10 - g, 3(10 - g) + 5g = 30 - 3g + 5g = 30 + 2g = 38, 2g = 8, g = 4, a = 6), (5) interpreting (6 liters apple, 4 liters grape), (6) verifying (6 + 4 = 10, 18 + 20 = 38). This matches choice A. Equations correctly represent totals. Errors: swapping costs or wrong solving like B. Setup: identify quantities, define, translate, solve, interpret, verify. Ensure non-negative sensible values.
A movie theater sells small popcorn for $4 and large popcorn for $6. On one night, they sold 50 popcorns total and collected $240. Let $s$ be the number of small popcorns and $l$ be the number of large popcorns. Which solution matches the situation?
Answer: $s=30,\ l=20$ This problem tests setting up and solving systems from real-world scenarios requiring two equations for two unknowns with context interpretation. The process involves: (1) defining variables (s = number of small popcorns, l = number of large popcorns), (2) writing total sold (s + l = 50), (3) writing revenue (4s + 6l = 240), (4) solving (s = 50 - l, 4(50 - l) + 6l = 200 + 2l = 240, 2l = 40, l = 20, s = 30), (5) interpreting (30 small, 20 large), (6) verifying (30 + 20 = 50, 120 + 120 = 240). This matches choice A. Equations correct for counts and earnings. Errors: reversing like B or extremes like C. Setup: identify, define, equations, solve, check positives.
A store sells notebooks and pens. A notebook costs $3 and a pen costs $1. A student buys 18 items total and spends $38. Let $n$ be the number of notebooks and $p$ be the number of pens. How many notebooks did the student buy?
Answer: 10 This question tests setting up and solving systems of equations from real-world scenarios requiring two equations for two unknowns with context interpretation. The process involves: (1) defining variables such as n for notebooks and p for pens, (2) writing n + p = 18 for total items, (3) writing 3n + p = 38 for total cost, (4) solving, (5) interpreting (number of notebooks), and (6) verifying. Solving correctly: subtract first from second: (3n + p) - (n + p) = 38 - 18, 2n = 20, n = 10, then p = 8. These satisfy: 10 + 8 = 18 and 3(10) + 8 = 30 + 8 = 38. Common errors include incorrect cost equation or not solving the system fully. To avoid errors: define clearly, translate costs and counts, use elimination or substitution, interpret the asked quantity, and verify.
The sum of two siblings' ages is 45 years, and the older sibling is 5 years older than the younger sibling. Let $o$ be the older sibling's age and $y$ be the younger sibling's age. What are their ages?
Answer: $o=25,\ y=20$ This question tests setting up and solving systems of equations from real-world scenarios requiring two equations for two unknowns with context interpretation. The process involves: (1) defining variables such as o for older sibling's age and y for younger's age in years, (2) writing the equation for sum: o + y = 45, (3) writing the equation for difference: o = y + 5, (4) solving by substitution, (5) interpreting the results, and (6) verifying. Solving correctly: substitute o = y + 5 into the first to get (y + 5) + y = 45, 2y + 5 = 45, 2y = 40, y = 20, o = 25. These satisfy both: 25 + 20 = 45 and 25 - 20 = 5. Common errors include reversing older and younger or misstating the difference, leading to pairs like o=20, y=25. To avoid errors: identify relationships clearly, define variables precisely, solve systematically, check for sensibility (ages positive), and verify both conditions.
Two friends start biking toward each other from towns that are 54 miles apart. One bikes at 12 mph and the other bikes at 6 mph. Let $t$ be the time in hours until they meet, and let $d$ be the distance (in miles) the 12 mph biker travels. Which system correctly models the situation, and what is $t$ ?
Answer: System: $d=12t$ and $d+6t=54$ ; $t=4.5$ This problem tests setting up and solving systems from real-world scenarios requiring two equations for two unknowns with context interpretation. The process involves: (1) defining variables with units (t = time in hours, d = distance in miles by 12 mph biker), (2) writing distance equation (d = 12t), (3) writing meeting equation (d + 6t = 54), (4) solving (substitute: 12t + 6t = 54, 18t = 54, t = 3), (5) interpreting (they meet after 3 hours), (6) verifying (d=36, 36+18=54). This matches choice A. System correct as it models distances summing to 54. Errors: wrong speeds or signs like C, wrong t like D. Setup: understand motion, define, equations from relations, solve, check. Consider relative speed for intuition.
A gym sells day passes and monthly passes. A day pass costs $6 and a monthly pass costs $30. On one day, the gym sold 40 total passes and collected $720. Let $d$ be the number of day passes and $m$ be the number of monthly passes. Which system represents the situation and what are $d$ and $m$ ?
Answer: System: $d+m=40$ and $6d+30m=720$ ; $d=20,\ m=20$ This question tests setting up and solving systems of equations from real-world scenarios requiring two equations for two unknowns with context interpretation. The process involves: (1) defining variables such as d for day passes and m for monthly passes, (2) writing d + m = 40 for total passes, (3) writing 6d + 30m = 720 for revenue, (4) solving, (5) interpreting, and (6) verifying. Solving correctly: from d = 40 - m, 6(40 - m) + 30m = 720, 240 - 6m + 30m = 720, 240 + 24m = 720, 24m = 480, m = 20, d = 20. These fit both equations. Common errors include swapping costs or totals, like using d + m = 720. To avoid errors: match constraints to equations carefully, solve step-by-step, ensure non-negative integers, and verify.
Two cars start 300 miles apart and drive toward each other. Car A drives 60 mph and Car B drives 40 mph. Let $t$ be the time in hours until they meet, and let $d$ be the distance Car A travels. Which system correctly models this situation?
Answer: $d=60t$ and $d+40t=300$ Since Car A travels at 60 mph, $d=60t$ . The cars start 300 miles apart and drive toward each other, so their distances together must add up to the full 300 miles: $d+40t=300$ . Choice B is wrong because the total distance between the cars is 300 miles, not 500. Choice C is wrong because $d$ represents Car A's distance, so its coefficient must be 60, not 40. Choice D is wrong because the cars are driving toward each other, so their distances add; subtraction would model the cars moving apart instead.
A science club needs 20 liters of a 30% salt solution. They have a 20% solution and a 50% solution. Let $x$ be liters of the 20% solution and $y$ be liters of the 50% solution. Which system correctly represents the situation?
Answer: $x+y=30$ and $0.2x+0.5y=20$ This problem tests setting up and solving systems from real-world scenarios requiring two equations for two unknowns with context interpretation. The process involves: (1) defining variables with units (x = liters of 20% solution, y = liters of 50% solution), (2) writing total volume (x + y = 20), (3) writing salt amount (0.2x + 0.5y = 6, since 30% of 20 is 6), (4) solving if needed, but question asks for system, (5) interpreting mixture, (6) verifying consistency. This matches choice A. System correct for volume and concentration. Errors: wrong total salt like B or swapped totals like C. Setup: understand mixing, define, balance quantities and amounts, check units.
Two cars start from the same point and travel in opposite directions. Car A travels 15 mph faster than Car B. After 2.5 hours, they are 275 miles apart. If Car A reduces its speed by 5 mph for the next hour while Car B maintains its speed, how far apart will they be after the additional hour?
Answer: The cars will be 380 miles apart after the additional hour Let b = Car B's speed and a = Car A's speed = b + 15. After 2.5 hours: 2.5b + 2.5(b + 15) = 275. This gives 2.5b + 2.5b + 37.5 = 275, so 5b = 237.5 and b = 47.5 mph. Car A's speed = 62.5 mph. In the next hour, Car A travels at 57.5 mph and Car B at 47.5 mph. Additional distance = 57.5 + 47.5 = 105 miles. Total distance apart = 275 + 105 = 380 miles. Choice A uses original speeds incorrectly. Choice B forgets Car A's speed reduction. Choice D adds the speed reduction instead of subtracting.
A student solves a system for a fundraiser and gets $(x,y)=(18,12)$ , where $x$ is the number of large water bottles sold and $y$ is the number of small water bottles sold. The student also knows 30 bottles were sold in total. What does $y=12$ mean in this context?
Answer: 12 small bottles were sold. The variables were defined so that $x$ represents large bottles and $y$ represents small bottles. Since $y=12$ , this means 12 small bottles were sold. Checking, $x+y=18+12=30$ , which matches the total number of bottles sold. Choice A incorrectly swaps which variable represents which type of bottle. Choices B and C misinterpret $y$ as a price or a total amount of money rather than a count of bottles.
A rectangle has perimeter 40 cm. Its length is 3 cm more than its width. Let $\ell$ be the length and $w$ be the width. What are the dimensions of the rectangle?
Answer: $w=8.5\text{ cm},\ \ell=11.5\text{ cm}$ This question tests setting up and solving systems of equations from real-world scenarios requiring two equations for two unknowns with context interpretation. The process involves: (1) defining variables such as w for width and ℓ for length in cm, (2) writing the perimeter equation: 2w + 2ℓ = 40 or w + ℓ = 20, (3) writing ℓ = w + 3, (4) solving by substitution, (5) interpreting, and (6) verifying. Solving correctly: (w + 3) + w = 20, 2w + 3 = 20, 2w = 17, w = 8.5, ℓ = 11.5. These satisfy: perimeter 2(8.5 + 11.5) = 40 and difference 3. Common errors include forgetting to double sides or misinterpreting 'more than.' To avoid errors: recall perimeter formula, define variables with units, translate relations accurately, solve, accept decimals if sensible, and verify.
The sum of two siblings' ages is 45 years, and the older sibling is 5 years older than the younger sibling. Let $x$ be the older sibling's age and $y$ be the younger sibling's age. Set up a system of equations and solve for $x$ and $y$ .
Answer: $x=25,\ y=20$ This problem tests setting up and solving systems from real-world scenarios requiring two equations for two unknowns with context interpretation. The process involves: (1) defining variables with units (let x = older sibling's age in years, y = younger sibling's age in years), (2) writing an equation from the sum constraint (x + y = 45), (3) writing an equation from the age difference (x = y + 5 or x - y = 5), (4) solving the system (substitute x = y + 5 into first: y + 5 + y = 45, 2y = 40, y = 20, x = 25), (5) interpreting (older is 25 years, younger is 20 years), and (6) verifying (25 + 20 = 45, 25 - 20 = 5). The correct solution is x=25, y=20, which matches choice C. The equations are correct as they capture the total age and the difference stated. Common errors include reversing the difference (y = x + 5) or arithmetic slips leading to other pairs like in A or D. To set up: identify constraints carefully, define variables with context, form equations accurately, solve and check for sensibility (ages positive and logical).
A student claims that $(x,y)=(4,7)$ is the solution to the system that models a fundraiser where $x$ is the number of large posters and $y$ is the number of small posters. Large posters cost $5 each, small posters cost $2 each, 11 posters were sold, and $34 was collected. Is the student correct?
Answer: Yes, because $4+7=11$ and $5(4)+2(7)=34$ . This question tests setting up and solving systems of equations from real-world scenarios requiring two equations for two unknowns with context interpretation, then verifying a proposed solution. The process involves: (1) defining variables such as x for large posters and y for small, (2) writing x + y = 11 for total posters, (3) writing 5x + 2y = 34 for revenue, (4) checking if (4,7) satisfies both, (5) interpreting if it's the solution, and (6) verifying calculations. Plugging in: 4 + 7 = 11 (true) and 5(4) + 2(7) = 20 + 14 = 34 (true), so yes. Common errors include miscalculation like 20 + 14 = 33 or ignoring one equation. To avoid errors: set up system first, plug values into both equations carefully, and confirm both are satisfied before concluding.

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