Question 132·200 Super-Hard SAT Math Questions·Algebra

Substitution means replacing letters with numbers. Choose $b=3$ and $c=4$. Then $b-1=2$, $c+1=5$, $bc=12$, and $b+c=7$ are all nonzero, so you can test every choice at this input.

In Desmos, type the two assignments and then the given equation with $a$ changed to $a_1$ and $=$ changed to $\sim$. The subscript makes $a_1$ a value for Desmos to find, and the regression matches the two sides. Under PARAMETERS, Desmos shows $a_1\approx2.8571$.

Type each choice's expression for $a$ on a separate line. Desmos shows about $2.8571$, $1.4286$, $1.6667$, and $1.4$, in that order. Only the first gives the value required by the original equation. So $a=\frac{2(b-1)(c+1)}{b+c}$. Choice A.

The two right-hand fractions have different denominators. Their common denominator is $(b-1)(c+1)$, and each numerator is $2$. Clear the denominators before isolating $a$. Multiply both sides by $\frac{(b-1)(c+1)}{2}$; each fraction on the right loses its denominator and its $2$:

The $+1$ and $-1$ cancel. Combine the remaining terms:

Multiply the whole equation by $a$:

The original denominators require $b-1\ne0$ and $c+1\ne0$. If $b+c=0$, the last equation would set their nonzero product equal to $0$, so dividing by $b+c$ is allowed. Divide both sides by $b+c$:

This expresses $a$ in terms of $b$ and $c$. Choice A.