AMC 10 · 2002 · #20

Grade 8 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

systems-of-equationsdifference-of-squares symmetry-argument ↑ Prerequisites: systems-of-equations

📏 Medium solution 💡 2 insights

Problem

Two equations tie together three real numbers a, b, and c: a - 7b + 8c = 4 and 8a + 4b - c = 7. Find the value of a² - b² + c².

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

Two equations with three letters cannot lock down a, b, and c separately, but they are enough to lock down two letters once the third is chosen. Tool #4 (Introduce a Variable) turns that freedom into a lever: keep a as a free stand-in and solve the two linear equations for b and c in terms of a. Then Tool #15 (Organize Information in More Ways) rewrites the target a² - b² + c² smartly — grouping c² - b² as a difference of squares (c-b)(c+b) — so the messy fractions collapse and the leftover a cancels. Tool #6 (Guess and Check) is the safety net: because the value is promised to be constant, plugging in one convenient choice of a letter must give the same number, so it doubles as a fast confirmation.

1STEP 1

Solve for b and c using a

Let a roam free and solve the pair for the other two: b = (12 - 13a)/5 and c = (13 - 12a)/5.

b = (12 - 13a)/5, c = (13 - 12a)/5

2STEP 2

Group c squared minus b squared

Read the target as a² + (c² - b²) and use c - b = (1 + a)/5, c + b = 5(1 - a) instead of squaring fractions.

c - b = (1 + a)/5, c + b = 5(1 - a)

3STEP 3

Multiply and cancel the a squared

Their product is (1 + a)(1 - a) = 1 - a², so a² - b² + c² = a² + (1 - a²) = 1, choice (B).

a² - b² + c² = a² + (1 - a²) = 1 (B)

Answer

The answer should not depend on which valid triple you pick, so test one. Let a = 0: the equations become -7b + 8c = 4 and 4b - c = 7; solving gives b = 12/5 and c = 13/5, and a² - b² + c² = 0 - 144/25 + 169/25 = 25/25 = 1. Try another: let c = 0, giving a - 7b = 4 and 8a + 4b = 7, which solve to a = 13/12, b = -5/12, and a² - b² + c² = 169/144 - 25/144 + 0 = 144/144 = 1. Both land on 1, matching choice (B).

💡Key takeaway

When two equations share more letters than they can pin down, let one letter float free, solve for the rest, and watch the leftover cancel — the answer that survives is the one that was fixed all along.

Solve for b and c using a
Group c squared minus b squared
Multiply and cancel the a squared

Solve for b and c using a

Group c squared minus b squared

Multiply and cancel the a squared

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