Q. 68 - Precalculus Enhanced with Graphing Utilities

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 $x-2y=-5$

containing the point $(0,4)$

Step-by-Step Solution

Answer

The equation of a line is $y=-2x+4$

Step 1. Given information

The perpendicular line is $x-2y=-5$

The point is $(0,4)$

Step 2. Finding the slope of line

Firstly, find the slope of a perpendicular line 

$x-2y=-5$

Solve for $y$

$$\begin{gathered} x-2y-x=-5-x \\ -2y=-x-5 \\ y=\frac{1}{2}x+\frac{5}{2} \end{gathered}$$

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 

$(x_1,y_1)=(0,4)$

Apply point-slope formula

$$\begin{gathered} y-4=-2(x-0) \\ y-4=-2x \\ y=-2x+4 \end{gathered}$$