Finding Perpendicular Lines: Equation of a Line Perpendicular to x - 5y -40

To find the equation of a line that is perpendicular to the line given by the equation x - 5y -40, we first need to determine the slope of the original line. Let's go through the steps in detail.

Step 1: Rewrite the Equation in Slope-Intercept Form

First, we need to rewrite the given equation in the slope-intercept form y mx b.

Rearrange the Given Equation

The given equation is:

x - 5y -40

To isolate y, we first move x to the right side:

-5y -x - 40

Now, divide every term by -5 to solve for y:

y frac{1}{5}x 8

From this, we can see that the slope m of the original line is frac{1}{5}.

Step 2: Find the Slope of the Perpendicular Line

The slope of a line that is perpendicular to another is the negative reciprocal of the original slope.

Calculate the Negative Reciprocal

Given that the slope of the original line is frac{1}{5}, the slope of the perpendicular line, m_{perpendicular}, is:

m_{perpendicular} -frac{1}{frac{1}{5}} -5

Step 3: Write the Equation of the Perpendicular Line

The equation of a line can be expressed in point-slope form as:

y - y_1 m(x - x_1)

where x_1, y_1 is a point on the line and m is the slope. If we choose a general point, such as the origin (0, 0), the equation becomes:

y - 0 -5(x - 0)

Simplifying, we get:

y -5x

We can also express this in standard form:

5x y 0

Therefore, one equation that represents a line perpendicular to x - 5y -40 is:

y -5x

Alternatively, in standard form: