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QuestionFind the equation of the line perpendicular to 2x=4y−82 x=4 y-8 that passes through the point (1,2)(1,2).

Studdy Solution

STEP 1

Assumptions1. The equation of the line we are given is x=4y−8x =4y -8. . The line we are looking for is perpendicular to the given line.
3. The line we are looking for passes through the point (1,)(1,).

STEP 2

First, we need to find the slope of the given line. We can do this by rearranging the equation into slope-intercept form, y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.
2x=4y−82x =4y -84y=2x+84y =2x +8y=12x+2y = \frac{1}{2}x +2

STEP 3

From the slope-intercept form of the given line, we can see that the slope of the given line is 12\frac{1}{2}.

STEP 4

The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. We can find this by taking the negative reciprocal of the slope of the given line.
mperpendicular=−1mgivenm_{perpendicular} = -\frac{1}{m_{given}}

STEP 5

Now, plug in the value for the slope of the given line to find the slope of the line perpendicular to it.
mperpendicular=−112m_{perpendicular} = -\frac{1}{\frac{1}{2}}

STEP 6

Calculate the slope of the line perpendicular to the given line.
mperpendicular=−112=−2m_{perpendicular} = -\frac{1}{\frac{1}{2}} = -2

STEP 7

Now that we have the slope of the line we are looking for, we can use the point-slope form of a line to find the equation of the line. The point-slope form is y−y1=m(x−x1)y - y1 = m(x - x1), where (x1,y1)(x1, y1) is a point on the line and mm is the slope of the line.

STEP 8

Plug in the values for the slope and the point (1,2)(1,2) into the point-slope form of a line.
y−2=−2(x−1)y -2 = -2(x -1)

STEP 9

implify the equation to find the equation of the line.
y−2=−2x+2y -2 = -2x +2y=−2x+4y = -2x +4The equation of the line that is perpendicular to the line defined by the equation 2x=4y−82x =4y -8 and goes through the point (,2)(,2) is y=−2x+4y = -2x +4.

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