Write an equation in slope intercept form for the line that passes through (8,5) and is perpendicular to the line described by y=-4x+7

Answer 1

First, we need to find the slope of the line described by the equation \( y = -4x + 7 \). In slope-intercept form \( y = mx + b \), the slope \( m \) is -4.

Since we want to find a line that is perpendicular to the given line, we need to find the negative reciprocal of the slope. The negative reciprocal of -4 is:

\[ m = \frac{1}{4} \]

Now we have the slope of the new line, which is \( \frac{1}{4} \). We will use the point-slope form of the equation of a line, which is given by:

\[ y - y_1 = m(x - x_1) \]

Here, \((x_1, y_1)\) is the point (8, 5) and \(m\) is \( \frac{1}{4} \).

Substituting these values into the equation gives us:

\[ y - 5 = \frac{1}{4}(x - 8) \]

Next, we will simplify this equation to get it into slope-intercept form \( y = mx + b \).

1. Distribute \( \frac{1}{4} \): \[ y - 5 = \frac{1}{4}x - 2 \]

2. Add 5 to both sides: \[ y = \frac{1}{4}x - 2 + 5 \] \[ y = \frac{1}{4}x + 3 \]

So, the equation of the line that passes through the point (8, 5) and is perpendicular to the line described by \( y = -4x + 7 \) is:

\[ \boxed{y = \frac{1}{4}x + 3} \]

Answer 2

By frac {1}{4} do you mean a feaction as one fourth?

Answer 3

Yes, that's correct! When I wrote \(\frac{1}{4}\), I was referring to the fraction that represents one fourth. In the context of the equation of the line, the slope \( \frac{1}{4} \) means that for every 4 units you move in the x-direction (to the right), the line moves up 1 unit in the y-direction.

So the final equation of the line, in slope-intercept form, is:

\[ \boxed{y = \frac{1}{4}x + 3} \]

This means that the slope of the line is one fourth.