Equations of Two Perpendicular Lines

In this page, we are going to find the equation of the line perpendicular to the equation:

Perp - 1

and goes through the point (2, -1).

We are going to put the equation of the line in Slope-Intercept Form which is:

Slope Intercept - 1

We’ll take this equation variable by variable, starting with the slope of the line, which is the m in the above equation. Two lines are considered perpendicular when the slopes of the two lines are opposite reciprocals of each other. Since the slope of the first line is 2/3, the slope of the other line would be –3/2.

Making our new equation:

perp 2  lines - 1.png

Now that we have our slope the last step would be to plug in the point we want the line to go through. In this case, the point is (2, -1). Our value is 2, and the y value is -1. Using these values the equation becomes:

perp 2  lines - 2.png

We can simplify the equation down by multiplying the 3/2 by 2, which is -3. Which gives us:

perp 2  lines - 3.png

Using the Addition Property of Equality, we can add 3 to both sides of the equation.

perp 2  lines - 4.png

-1 plus 3 is 2, and -3 plus 3 is 0, which will give us the value of b.

perp 2  lines - 5.png

Now that we have the value of b we can plug it into the slope-intercept equation, which yields the following equation:

perp 2  lines - 6.png

We can check our work by graphing the two lines, as seen below:

perp 2  lines - 7.png

In the above figure, the blue line is the original graph, and the red line is one we computed. The two lines do intersect each other, and the angle they form is 90°, making the two lines perpendicular to each other. And with the red line going through the point (2, -1)  it confirms that our answer is correct.