Write an equation in slope intercept form perpendicular to:
-2x+8y=16 and goes through (4,5)
Slope of given line = 2/8 = 1/4
so the slope of the new line = -4
y = -4x + b , but (4,5) is on it, so ...
5 = -4(4) + b
b = 21
y = -4x + 21
To determine an equation in slope-intercept form that is perpendicular to a given line and passes through a specific point, you need to follow a few steps:
Step 1: Find the slope of the given line.
Step 2: Determine the negative reciprocal of the slope found in step 1. This will be the slope of the line perpendicular to the given line.
Step 3: Use the point-slope form of a line to write the equation, substituting the values of the point and the slope found in step 2.
Step 4: Rewrite the equation in slope-intercept form.
Let's go through these steps for the given line -2x + 8y = 16 and the point (4,5):
Step 1: Find the slope of the given line.
To find the slope, let's rearrange the equation into the form y = mx + b, where m represents the slope.
-2x + 8y = 16
8y = 2x + 16
y = (2/8)x + 2/8
y = (1/4)x + 1/4
From the equation, we can see that the slope of the given line is 1/4.
Step 2: Determine the negative reciprocal of the slope.
The perpendicular line's slope will be the negative reciprocal of the given line's slope. So, the perpendicular line will have a slope of -4/1 or -4.
Step 3: Use the point-slope form of a line to write the equation.
The point-slope form of a line is: y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope of the line.
Substituting the given point (4,5) and the slope -4 into the equation, we get:
y - 5 = -4(x - 4)
Step 4: Rewrite the equation in slope-intercept form.
Let's simplify the equation by distributing and rearranging terms:
y - 5 = -4x + 16
y = -4x + 16 + 5
y = -4x + 21
Therefore, the equation of the line perpendicular to -2x + 8y = 16 and passing through the point (4,5) in slope-intercept form is y = -4x + 21.