AMC 10 · 2019 · #4

Grade 6 algebra

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

sequences-arithmeticcoordinate-geometrylinear-equations-two-varpattern-recognition identify-subproblemspattern-recognition ↑ Prerequisites: sequences-arithmeticlinear-equations-two-var

📏 Medium solution 💡 3 insights

Problem

Consider every line of the form a x + b y = c where a, b, c are three numbers in arithmetic progression (each next term differs from the previous by the same amount). All such lines pass through one common point. Find the coordinates of that point.

Pick an answer.

(A)

(-1,2)

(B)

(0,1)

(C)

(1,-2)

(D)

(1,0)

(E)

(1,2)

AMC 10 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Eliminate Possibilities

Tool #9 (Easier Problem): we don't need to handle every AP — pick the two simplest ones, say (a, b, c) = (1, 2, 3) and (1, 3, 5). These give two specific lines x + 2y = 3 and x + 3y = 5. Their common point must include the answer. Tool #3 (Eliminate) + Tool #6 (Guess and Check): plug each of the five answer choices into both lines; the only point that satisfies BOTH equations is the answer. No algebra needed.

1STEP 1

Pick two simple APs: (1, 2, 3) gives x + 2y = 3, and (1, 3, 5) gives x + 3y = 5. The point must satisfy both.

Line 1: x + 2y = 3 Line 2: x + 3y = 5

2STEP 2

Plug each choice into x + 2y = 3: only (-1, 2) works (-1 + 4 = 3); (0,1), (1,-2), (1,0), (1,2) all fail.

In x + 2y = 3: only (-1, 2) works (-1 + 4 = 3)

3STEP 3

Check (-1, 2) on the second line x + 3y = 5: -1 + 6 = 5 ✓, so it lies on both lines.

In x + 3y = 5: (-1) + 3(2) = -1 + 6 = 5 ✓

4STEP 4

Test more APs: (2, 5, 8) gives 2(-1)+5(2)=8 ✓, (3, 2, 1) gives 3(-1)+2(2)=1 ✓ — (-1, 2) works for every AP.

(-1, 2): 2(-1) + 5(2) = 8 ✓; 3(-1) + 2(2) = 1 ✓ → (A)

Answer

(-1,2)

Sanity: an AP gives c = 2b - a. Plug (x, y) = (-1, 2) into a x + b y: a(-1) + b(2) = -a + 2b = 2b - a = c ✓. So for ANY arithmetic-progression triple (a, b, c), the line ax + by = c passes through (-1, 2). The answer holds universally — not just for our test cases.

💡Key takeaway

This AMC 10 problem only needs Grade 6 expression-evaluation you already know — pick two simple APs (1, 2, 3) and (1, 3, 5), plug each answer-choice point into both lines, and only (-1, 2) survives. Answer: (A)!

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