AMC 10 · 2005 · #2
Grade 7 arithmeticPick an answer.
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.
Solve the first equation for x
The first equation has only one unknown, giving -2.
An equation with a single unknown pins that unknown to one number — undo the +7 and the ×2 to reveal it.
7.EE.B.4Identify SubproblemsUse "same solution" to link the equations
Same solution means that value must fit the second equation too.
"Same solution" is a bridge: the number you found on one side must fit the other side too.
Saying the two equations share a solution means the number found from one must also satisfy the other.
▸ Why?
If the unknown equals a particular number, anything true of the unknown is true of that number.
▸ Why?
Undoing the same operations on both sides keeps the equation true while stripping the unknown bare.
Solve for b
Solving with attention to the sign gives -4, choice (B).
Dividing a positive by a negative flips the sign, so the coefficient lands below zero.
7.NS.A.2Work BackwardsWhen 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