Question 163·200 Super-Hard SAT Math Questions·Algebra

Infinitely many solutions means both equations describe the same line. A multiplier is the number that turns one equation into the other. The same multiplier must scale every term, including the constant. Call it $m$ and match the known $x$ coefficients: $m\left(\frac{11}{12}\right)=\frac{35}{32}$. You don't need to find $a$ or $c$.

Divide the second $x$ coefficient by the first. Type $\frac{35}{32}/\frac{11}{12}$ in Desmos. It shows about $1.19318$; tap the fraction button to read $m=\frac{105}{88}$.

The first equation ends in $b$, so multiplying it by $m=\frac{105}{88}$ must give the second equation's right-hand side: $4d=\frac{105}{88}b$. Keep the $4$ attached to $d$; the second equation doesn't end in $d$ alone.

Divide by $d$ to bring out $b/d$. The quotient in the question requires $d\ne0$: $4=\frac{105}{88}\frac{b}{d}$.

Let $x_1$ stand for $b/d$. Type $\frac{105}{88}x_1\sim4$: Desmos uses $\sim$ to find the value of $x_1$ that matches both sides. Under PARAMETERS, it shows $x_1\approx3.35238$. Type $x_1$ on the next line and tap its fraction button to get $\frac{352}{105}$. So $\frac{b}{d}=\frac{352}{105}$. Choice C.