Solve a pair of linear equations: Twice the first number plus thrice the second equals 100, and thrice the first plus twice the second equals 120. Which is the larger number?
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A32
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B12
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C14
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D35
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E28
Answer
Correct Answer: 32
Explanation
Introduction / Context:This is a direct two-equation, two-unknowns problem. Solve the linear system to find both numbers, then identify the larger one. Elimination using aligned coefficients makes the arithmetic quick and clean.
Given Data / Assumptions:
- 2x + 3y = 100
- 3x + 2y = 120
Concept / Approach:Eliminate one variable by forming equal coefficients on x or y. Subtract the resulting equations to find the other variable, then back-substitute to get the first. Compare the values to decide which is larger.
Step-by-Step Solution:
Multiply the first equation by 3: 6x + 9y = 300 Multiply the second equation by 2: 6x + 4y = 240 Subtract: (6x + 9y) − (6x + 4y) = 60 ⇒ 5y = 60 ⇒ y = 12 Substitute into 3x + 2y = 120 ⇒ 3x + 24 = 120 ⇒ 3x = 96 ⇒ x = 32 Larger number = 32Verification / Alternative check:Check in the first equation: 2*32 + 3*12 = 64 + 36 = 100 ✔; second: 3*32 + 2*12 = 96 + 24 = 120 ✔.
Why Other Options Are Wrong:12 and 14 are the smaller or unrelated values; 35 and 28 do not satisfy both equations simultaneously.
Common Pitfalls:Arithmetic slips when scaling equations or subtracting can flip signs and produce wrong values; always verify in both original equations.
Final Answer:32