Question

Find an equation for the line with the given properties. Express your answer using either the general form or the slope-intercept form of the equation of a line, whichever you prefer.

Perpendicular to the line $y=\frac{1}{2}x+4$; containing the point$(1,-2)$.

Short answer

The equation of the line is$y=-2x$.

Step-by-step solution

Step 1. Given information

The perpendicular line is $y=\frac{1}{2}x+4$.

The point is$(1,-2)$

Step 2. Finding the slope of line

Firstly, find the slope of perpendicular line

$y=\frac{1}{2}x+4$

Compare with slope-intercept formula

$y=mx+b$

So, slope is $\frac{1}{2}$

Since slope of perpendicular line is always negative of inverse of the other line slope.

So, slope is

$$\begin{gathered} m=-\frac{1}{{\displaystyle \frac{1}{2}}} \\ m=-2 \end{gathered}$$

Step 3. Finding the equation of line

The slope of line is

$m=-2$

The point is

$(x_1,y_1)=(1,-2)$

Apply point-slope formula

$y-y_1=m(x-x_1)$

Plug values

$$\begin{gathered} y+2=-2(x-1) \\ y+2=-2x+2 \\ y+2-2=-2x+2-2 \\ y=-2x \end{gathered}$$