SAT Question 9d0396d4: Linear Equations In Two Variables answer and explanation with Desmos solution

Choice C is correct. It's given from the graph that the points $\left(0,7\right)$ and $\left(8,0\right)$ lie on the line. For two points on a line, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, the slope of the line can be calculated using the slope formula $m=\frac{y_2-y_1}{x_2-x_1}$. Substituting $\left(0,7\right)$ for $\left(x_1,y_1\right)$ and $\left(8,0\right)$ for $\left(x_2,y_2\right)$ in this formula, the slope of the line can be calculated as $m=\frac{0-7}{8-0}$, or $m=-\frac{7}{8}$. It's also given that the point $\left(d,4\right)$ lies on the line. Substituting $\left(d,4\right)$ for $\left(x_1,y_1\right)$, $\left(8,0\right)$ for $\left(x_2,y_2\right)$, and $-\frac{7}{8}$ for $m$ in the slope formula yields $-\frac{7}{8}=\frac{0-4}{8-d}$, or $-\frac{7}{8}=\frac{-4}{8-d}$. Multiplying both sides of this equation by $8-d$ yields $-\frac{7}{8}\left(8-d\right)=-4$. Expanding the left-hand side of this equation yields $-7+\frac{7}{8}d=-4$. Adding $7$ to both sides of this equation yields $\frac{7}{8}d=3$. Multiplying both sides of this equation by $\frac{8}{7}$ yields $d=\frac{24}{7}$. Thus, the value of $d$ is $\frac{24}{7}$.