Find an equation for the line perpendicular to $4 x+8 y=16$ and goes through the point $(8,-1)$. Write your answer in the form $y=m x+b$. \[ y= \] Submit Question

Step 1 ：Rewrite the given equation $4x + 8y = 16$ in the form $y = mx + b$ to find the slope of the given line. This gives us $y = -\frac{1}{2}x + 2$ so the slope of the given line is $-\frac{1}{2}$.

Step 2 ：The slope of a line perpendicular to the given line is the negative reciprocal of the slope of the given line. So, the slope of the perpendicular line is $-\frac{1}{-\frac{1}{2}} = 2$.

Step 3 ：Use the point-slope form of a line, $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the point the line goes through, to find the equation of the line. Substituting $m = 2$, $x_1 = 8$, and $y_1 = -1$ gives us $y - (-1) = 2(x - 8)$.

Step 4 ：Simplify the equation to get it in the form $y = mx + b$. This gives us $y = 2x - 17$.

Step 5 ：Final Answer: The equation of the line perpendicular to $4 x+8 y=16$ and goes through the point $(8,-1)$ is \(\boxed{y=2x-17}\).