sat-dog

The correct answer is $33$. It’s given in the table that the coordinates of two points on a line in the xy-plane are $(k,13)$ and $(k+7,-15)$. The y-intercept is another point on the line. The slope computed using any pair of points from the line will be the same. The slope of a line, $m$, between any two points, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, on the line can be calculated using the slope formula, $m=\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}$. It follows that the slope of the line with the given points from the table, $(k,13)$ and $(k+7,-15)$, is $m=\frac{-15-13}{k+7-k}$, which is equivalent to $m=\frac{-28}{7}$, or $m=-4$. It's given that the y-intercept of the line is $(k-5,b)$. Substituting $-4$ for $m$ and the coordinates of the points $(k-5,b)$ and $(k,13)$ into the slope formula yields $-4=\frac{13-b}{k-\left(k-5\right)}$, which is equivalent to $-4=\frac{13-b}{k-k+5}$, or $-4=\frac{13-b}{5}$. Multiplying both sides of this equation by $5$ yields $-20=13-b$. Subtracting $13$ from both sides of this equation yields $-33=-b$. Dividing both sides of this equation by $-1$ yields $b=33$. Therefore, the value of $b$ is $33$.