SAT Question d4572f55: Linear Functions answer and explanation with Desmos solution

Choice D is correct. It’s given that for a linear relationship between $x$ and $y$, the table gives three values of $x$ and their corresponding values of $y$. A linear equation can be written in the form $y=mx+b$, where $m$ is the slope and $(0,b)$ is the y-intercept of the graph of the equation in the xy-plane. The slope of a line that passes through the points $(x_1,y_1)$ and $(x_2,y_2)$ can be calculated using the formula $m=\frac{y_2-y_1}{x_2-x_1}$. Substituting $(-34,t)$ and $(-17,t+23)$ for $(x_1,y_1)$ and $(x_2,y_2)$, respectively, in the formula $m=\frac{y_2-y_1}{x_2-x_1}$ yields $m=\frac{(t+23)-t}{(-17)-(-34)}$, which is equivalent to $m=\frac{t+23-t}{-17+34}$, or $m=\frac{23}{17}$. Based on the table, the graph of this relationship passes through the point $(0,t+46)$. It follows that $b=t+46$. Substituting $\frac{23}{17}$ for $m$ and $t+46$ for $b$ in the equation $y=mx+b$ yields $y=\frac{23}{17}x+t+46$. Therefore, the equation $y=\frac{23}{17}x+t+46$ represents this relationship.