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 (l = length in cm, w = width in cm), (2) writing perimeter equation (2l + 2w = 40 or l + w = 20), (3) writing relation (l = w + 3), (4) solving (w + 3 + w = 20, 2w + 3 = 20, 2w = 17, w = 8.5, l = 11.5), (5) interpreting (length 11.5 cm, width 8.5 cm), (6) verifying (11.5 + 8.5 = 20, times 2 = 40; difference 3). This matches choice A. Equations correct for perimeter and difference. Errors: ignoring factor of 2 or reversing like D. Setup: recall formulas, define, translate, solve, verify positives.

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.

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).

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.

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.

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.

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 (n = number of notebooks, p = number of pens), (2) writing total items (n + p = 9), (3) writing total cost (2n + p = 14), (4) solving (subtract: n = 5, then p = 4), (5) interpreting (5 notebooks, 4 pens), (6) verifying (5 + 4 = 9, 10 + 4 = 14). This matches choice B. Equations correct for quantities and costs. Errors: swapping costs or wrong counts like A. Setup: identify totals, define, equations, solve, check integers. Ensure practical sense.

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.

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 = large bottles, y = small bottles), (2) writing total bottles (x + y = 30), (3) writing revenue if needed (but here solution given), (4) interpreting the given solution (x=18 large, y=12 small), (5) verifying it fits (18+12=30), and (6) explaining y=12 as 12 small bottles sold. The value y=12 means 12 small bottles were sold. This interpretation is correct as it directly ties to the variable definition and context. Common errors include misinterpreting variables or ignoring the total. To set up correctly: understand given solution, relate to context, verify consistency, and explain clearly. Avoid errors like confusing x and y or adding extraneous details.

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 t = time in hours, d = distance Car A travels in miles), (2) writing the equation for Car A's distance (d = 60t), (3) writing the equation for total distance (d + 40t = 300), (4) solving by substitution (60t + 40t = 300, 100t = 300, t = 3), (5) interpreting (they meet after 3 hours), and (6) verifying distances add to 300. The correct system is d = 60t and d + 40t = 300, yielding t = 3 hours. These equations are correct because they account for each car's speed and the closing distance. Common errors include switching speeds or using subtraction instead of addition, leading to incorrect times like t=5. To set up correctly: define variables based on the problem's request, model directions accurately, solve consistently, interpret in context, and verify. Avoid errors like miscalculating the solution or ignoring relative motion.