Question 72·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

No Desmos needed. A numerical check can't prove a formula works for every $a$ and $b$. The outside multiplier on $c+4$ is $27$. Divide both sides of the given equation by $27$:

Undo operations on $c$, not the cube on $2a-5b$.

The remaining operation on $c$ is adding $4$. Subtract $4$ from both sides:

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

Set $a=5$ and $b=1$ in Desmos. It shows those two distinct values. A test can rule out a proposed formula, though one test can't prove it works for every $a$ and $b$.

Type the given equation with $c_1$ in place of $c$ and $\sim$ in place of $=$. The $\sim$ tells Desmos to fit the unknown value. Under PARAMETERS, it shows $c_1\approx0.62963$, which is distinct from $5$ and $1$.

Enter the four proposed right-hand sides in order. Desmos prints approximately $-2.333$, $371$, $8.630$, and $0.630$. Only the fourth result matches $c_1\approx0.630$. This checks the choice for these test values; the hand rearrangement proves the formula in general. Choice D.