SAT Question 768f3b7c: Linear Equations In Two Variables answer and explanation with Desmos solution

Choice B is correct. The linear relationship between $x$ and $y$ can be represented by an equation of the form $y-y_1=m\left(x-x_1\right)$, where $m$ is the slope of the graph of the equation in the xy-plane and $\left(x_1,y_1\right)$ is a point on the graph. The slope of a line can be found using two points on the line and the slope formula $m=\frac{y_2-y_1}{x_2-x_1}$. Each value of $x$ and its corresponding value of $y$ in the table can be represented by a point $\left(x,y\right)$. Substituting the points $\left(-s,21\right)$ and $\left(s,15\right)$ for $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, respectively, in the slope formula yields $m=\frac{15-21}{s-\left(-s\right)}$, which gives $m=\frac{-6}{2s}$, or $m=-\frac{3}{s}$. Substituting $-\frac{3}{s}$ for $m$ and the point $\left(s,15\right)$ for $\left(x_1,y_1\right)$ in the equation $y-y_1=m\left(x-x_1\right)$ yields $y-15=-\frac{3}{s}\left(x-s\right)$. Distributing $-\frac{3}{s}$ on the right-hand side of this equation yields $y-15=-\frac{3x}{s}+3$. Adding $15$ to each side of this equation yields $y=-\frac{3x}{s}+18$. Multiplying each side of this equation by $s$ yields $sy=-3x+18s$. Adding $3x$ to each side of this equation yields $3x+sy=18s$. Therefore, the equation $3x+sy=18s$ represents this relationship.