Math

QuestionFind the equation of the line through (4,1)(4,1) that is perpendicular to y=43x+4y=-\frac{4}{3}x+4.

Studdy Solution

STEP 1

Assumptions1. We are given a point (4,1)(4,1) through which the line passes. . We are given a line y=43x+4y=-\frac{4}{3}x+4 to which the line is perpendicular.
3. The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line.

STEP 2

First, we need to find the slope of the given line. The slope of a line in the form y=mx+by=mx+b is mm.
The given line is y=4x+4y=-\frac{4}{}x+4, so the slope of the given line is 4-\frac{4}{}.

STEP 3

Next, we need to find the slope of the line that is perpendicular to the given line. This is the negative reciprocal of the slope of the given line.
The negative reciprocal of 3-\frac{}{3} is 3\frac{3}{}.
So, the slope of the line that is perpendicular to the given line is 3\frac{3}{}.

STEP 4

Now that we have the slope of the line, we can use the point-slope form of a line to find the equation of the line. The point-slope form of a line is yy1=m(xx1)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 5

Plug in the values for the slope and the point into the point-slope form of a line to find the equation of the line.
y1=34(x4)y-1=\frac{3}{4}(x-4)

STEP 6

implify the equation of the line.
y1=34x3y-1=\frac{3}{4}x-3

STEP 7

olve the equation for yy.
y=34x3+1y=\frac{3}{4}x-3+1

STEP 8

implify the equation.
y=34x2y=\frac{3}{4}x-2The equation of the line that passes through the point (4,1)(4,1) and is perpendicular to the line y=43x+4y=-\frac{4}{3}x+4 is y=34x2y=\frac{3}{4}x-2.

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