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2005 AMC 10A Problem 3

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2005 AMC 10A Problem 3AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 1 of 5 · about 1 min · position P1-10

The equations $2x + 7 = 3$ and $bx - 10 = -2$ have the same solution $x$ . What is the value of $b$ ?

A)
$-8$
B)
$-4$
C)
$-2$
D)
$4$
E)
$8$

Answer

Key insight

Solve the equation without b to get x = -2, then substitute that x into the second equation and solve for b.

Solution

The first equation involves only $x$ : $2x + 7 = 3$ gives $2x = -4$ , so $x = -2$ .

That same $x$ satisfies the second equation. Substituting:
$b(-2) - 10 = -2 \quad\Longrightarrow\quad -2b = 8 \quad\Longrightarrow\quad b = -4.$

The answer is $\boxed{\textbf{(B)}\ -4}$ .

Why this works

When two equations share a solution, solve the one you can solve, then feed the result into the other. The unknown $b$ is a coefficient, but once $x$ is a specific number, the second equation is just a linear equation in $b$ .

Alternative approach

Check choices directly: with $x = -2$ the second equation reads $-2b = 8$ ; only $b = -4$ makes that true.

The trap

A sign slip when substituting x = -2 (writing 2b - 10 = -2 and getting b = 4, choice (D)).

Common mistakes

A sign slip when substituting x = -2 (writing 2b - 10 = -2 and getting b = 4, choice (D)).
Solving $2x + 7 = 3$ as $x = 2$ (forgetting the sign of $-4/2$ ) and getting $b = 4$ .

Techniques

Set up the equation/formula and compute; no special trick needed