AMC 10 · 2017 · #3
Grade 7 algebraPick an answer.
Tool #4 (Introduce a Variable): the given equation is a single relation between the two unknowns x and y. Clearing the fraction turns it into a plain linear equation, and solving that equation reveals how x and y are tied together. Once that tie is known, the target expression collapses to a number. Tool #6 (Guess and Check) lets us pick a concrete pair (x,y) that fits and read the answer directly. Tool #3 (Eliminate Possibilities) matches the result to the five choices.
Clear the fraction
Multiplying by the denominator clears the fraction.
Multiplying away the denominator turns a fraction equation into an ordinary one.
Multiplying away the denominator turns a fraction equation into an ordinary one.
▸ Why?
Multiplying by a denominator undoes the division, so the fraction disappears.
▸ Why?
Doing the same thing to both sides keeps the equation true, so nothing is lost in the clearing.
Find how x and y relate
Tidying says the two numbers are equal.
The whole condition just says x and y are equal.
6.EE.A.3Introduce A VariableSubstitute into the target
Substituting makes the letters cancel completely.
With x=y both pieces are multiples of y, so y divides out.
7.EE.A.1Guess And CheckMatch the choice
What remains is 2.
The single value we computed is the one we report.
6.NS.A.1Eliminate PossibilitiesClear the fraction first: (3x+y)/(x-3y)=-2 means 3x+y=-2(x-3y), which boils down to x=y, and then (x+3y)/(3x-y)=4y/2y=2, choice (D).
- Clear the fraction
- Find how x and y relate
- Substitute into the target
- Match the choice