Question

Write an equation for line L in point-slope form and

slope-intercept form.
L is perpendicular to y = 4x.

Answer 1

Point-slope form:
`y-4=-(1)/(4) (x-1)`

Slope-intercept form:
`y=-(1)/(4)x+(17)/(4)`

Explanation:

Pre-Solving

We are given that one line has the equation y = 4x.
We also know that line L is perpendicular to y = 4x, and that both lines passes through (1,4).

We want to write the equation of line L in point-slope form and slope-intercept form.

Point-slope form is
`y-y_1=m(x-x_1)` where m is the slope and
`(x_1,y_1)` is a point.

Slope-intercept form is y=mx+b, where m is the slope and b is the value of y at the y-intercept.

Also recall that perpendicular lines have slopes that multiply to equal -1.

Solving

Slope

First, we need to get the slope of y=4x.

Notice how the line is in slope-intercept form, and that 4 is in the place of where m is in mx.

This means that the slope of this line is 4.

Now, we need to find the slope of the perpendicular line.

As already stated, perpendicular lines multiply to equal -1.

So, we can say:

4m = -1

Divide both sides by 4.

`m =- (1)/(4)`

This is the slope of line L.

Point-Slope Form

Let's find the equation of line L in point-slope form.

As we now found the slope of this line, we can substitute that value of that line into the formula.

We get:

`y-y_1=-(1)/(4) (x-x_1)`

Now, substitute 1 as
`x_1` and 4 as
`y_1`.

`y-4=-(1)/(4) (x-1)`

Slope-Intercept Form

Now, let's find the equation of line L in slope-intercept form.

We can take take the point-slope form that we found earlier to get into slope-intercept form. We can notice that in slope-intercept form, we have only y on one side.

We can start by distributing -1/4 to both x and -1 on the right side.

`y-4=-(1)/(4)x+(1)/(4)`

Now, add 4 to both sides.

`y=-(1)/(4)x+(17)/(4)`