2005 AMC 10A Problem 3

P1-10~1 minPrint

Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

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

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Key insight

The first equation has no b, so solve it to get x = -2. Put that x into the second equation and solve for b.

Solution

Only the first equation can be solved right away, because it has just one unknown. Start there.

1. Find $x$ . From $2x + 7 = 3$ , subtract $7$ from both sides: $2x = -4$ . Divide by $2$ : $x = -2$ .
2. Use the same $x$ in the second equation. The problem says both equations have the same solution, so $x = -2$ must also make $bx - 10 = -2$ true:
$b \cdot (-2) - 10 = -2.$
3. Solve for $b$ . Add $10$ to both sides: $-2b = 8$ . Divide by $-2$ : $b = -4$ .

Check: with $b = -4$ and $x = -2$ , we get $(-4)(-2) - 10 = 8 - 10 = -2$ . Correct.

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

Why this works

When two equations share an answer, solve the easier one first and carry that answer into the other one. Once $x$ is a plain number, the second equation has only one unknown left, $b$ , and becomes an ordinary one-step equation.

Alternative approach

Try the choices. With $x = -2$ , the second equation becomes $-2b - 10 = -2$ . Testing $b = -4$ gives $8 - 10 = -2$ , which works; the other choices do not.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.4)