AMC 10 · 2005 · #3

Grade 7 arithmetic

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

linear-equations-one-varsystems-of-equations identify-subproblemswork-backwards ↑ Prerequisites: linear-equations-one-var

📏 Short solution 💡 1 insight

Problem

Two equations,

2x+7=3

and

bx-10=-2

, have the same solution

x

. That means one particular number for

x

makes both equations true at once. What is the value of

b

Pick an answer.

(A)

-8

(B)

-4

(C)

-2

(D)

(E)

AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The phrase "same solution" splits the task into two clean subproblems, so Tool #7 (Identify Subproblems) leads: first find the single x that solves the self-contained first equation, then feed that number into the second equation. Tool #11 (Work Backwards) handles the second stage — the second equation gives the end result -2, and from it we reverse-engineer the coefficient b. Tool #3 (Eliminate Possibilities) guards the sign traps: dropping a minus sign turns the answer into (D) 4, and copying x by mistake gives (C) -2.

1STEP 1

Solve the first equation for x

Only x appears in $2x+7=3$ : subtract 7 to get $2x=-4$ , then divide by 2, so $x=-2$ .

2x+7=3 → 2x=-4 → x=-2

2STEP 2

Use "same solution" to link the equations

Same solution means $x=-2$ must fit $bx-10=-2$ too, leaving $-2b-10=-2$ with b as the only unknown.

b(-2)-10=-2 → -2b-10=-2

3STEP 3

Solve for b

Add 10 to both sides: $-2b=8$ . Divide by $-2$ and the sign flips, so $b=-4$ — choice (B).

-2b-10=-2 → -2b=8 → b=-4 → (B)

Answer

-4

Check by putting both numbers back in: with x=-2 and b=-4, the second equation reads (-4)(-2)-10=8-10=-2, which matches the required -2 exactly. The first equation also checks: 2(-2)+7=-4+7=3. Both hold, so b=-4 is right. The trap answers make sense as near-misses: (D) 4 is what you get from a sign slip when dividing by -2, and (C) -2 is just the value of x copied over by mistake.

💡Key takeaway

When two equations share a solution, solve the one that has a single unknown first, then plug that number into the other to unlock what is left.

Solve the first equation for x
Use "same solution" to link the equations
Solve for b

Solve the first equation for x

Use "same solution" to link the equations

Solve for b

More like this