AMC 10 · 2005 · #3
Grade 7 arithmeticThe equations 2x+7=3 and bx−10=−2 have the same solution x. What is the value of b?
Pick an answer.
AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Two equations, $2x+7=3$ and $bx-10=-2$, are said to have the same solution $x$. That means one particular number for $x$ makes both equations true at once. Find the value of $b$ that makes this happen.
Givens: First equation: $2x+7=3$; Second equation: $bx-10=-2$; Both equations are true for the same value of $x$; Answer choices: (A) $-8$, (B) $-4$, (C) $-2$, (D) $4$, (E) $8$
Unknowns: The value of $b$ (the coefficient in the second equation)
Understand
Restated: Two equations, $2x+7=3$ and $bx-10=-2$, are said to have the same solution $x$. That means one particular number for $x$ makes both equations true at once. Find the value of $b$ that makes this happen.
Givens: First equation: $2x+7=3$; Second equation: $bx-10=-2$; Both equations are true for the same value of $x$; Answer choices: (A) $-8$, (B) $-4$, (C) $-2$, (D) $4$, (E) $8$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #11 Work Backwards, #3 Eliminate Possibilities
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) $-2$.
Execute — Answer: B
7.EE.B.4 Step 1 Solve the first equation for x
- The first equation has only $x$ in it, so solve it by itself.
- Subtract $7$ from both sides: $2x=3-7=-4$.
- Then divide both sides by $2$: $x=-2$.
- This is the one value of $x$ the problem is talking about.
💡 An equation with a single unknown pins that unknown to one number — undo the $+7$ and the $\times2$ to reveal it.
7.EE.B.4 Step 2 Use "same solution" to link the equations
- Because both equations share the same solution, the value $x=-2$ must also make the second equation true.
- Replace $x$ with $-2$ in $bx-10=-2$ to get an equation whose only unknown is now $b$: $b(-2)-10=-2$, that is $-2b-10=-2$.
💡 "Same solution" is a bridge: the number you found on one side must fit the other side too.
7.NS.A.2 Step 3 Solve for b
- Now solve $-2b-10=-2$ for $b$.
- Add $10$ to both sides: $-2b=8$.
- Divide both sides by $-2$: $b=-4$.
- Watch the sign — dividing $8$ by $-2$ gives a negative, so $b=-4$, not $4$.
- That makes the answer (B).
💡 Dividing a positive by a negative flips the sign, so the coefficient lands below zero.
7.EE.B.4 The first equation has only $x$ in it, so solve it by itself. Subtract $7$ from 7.EE.B.4 Because both equations share the same solution, the value $x=-2$ must also make 7.NS.A.2 Now solve $-2b-10=-2$ for $b$. Add $10$ to both sides: $-2b=8$. Divide both side Review
Reasonableness: 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.
Alternative: Skip solving the second equation for $b$ directly and instead isolate the $bx$ term first. From $bx-10=-2$ add $10$ to get $bx=8$. Since $x=-2$, this is $-2b=8$, so $b=8\div(-2)=-4$. Same landing at (B), just grouping the constant before substituting.
CCSS standards used (min grade 7)
7.EE.B.4Use variables to represent quantities and construct simple equations and inequalities (Solving the two-step equation $2x+7=3$ of the form $px+q=r$ to find $x=-2$, and rewriting the second equation with that value substituted in.)7.NS.A.2Apply and extend understanding of multiplication and division of rational numbers (Dividing $8$ by $-2$ to isolate $b$, correctly tracking the sign to get $b=-4$ rather than $4$.)
⭐ 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.
⭐ 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.
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