QuestionFind the equation of a line through $(-4,4)$ that is perpendicular to $y=\frac{1}{2}x+2$ .

Studdy Solution

STEP 1

Assumptions1. We know the point through which the line passes $(-4,4)$ . We know the line to which our line is perpendicular $y=\frac{1}{} x+$
3. The slope of the 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 $y=mx+b$ is $m$ .So, for the line $y=\frac{1}{2} x+2$ , the slope is $\frac{1}{2}$ .

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 $\frac{1}{2}$ is $-2$ .
So, the slope of the line we are trying to find is $-2$ .

STEP 4

Now that we have the slope of the line, we can use the point-slope form of a line to write the equation of the line. The point-slope form is $y-y1=m(x-x1)$ , where $(x1,y1)$ is a point on the line and $m$ is the slope.

STEP 5

Plug in the values for the slope and the point to write the equation of the line.
$y-4=-2(x+4)$

STEP 6

implify the equation by distributing the $-2$ on the right side of the equation.
$y-4=-2x-8$

STEP 7

Finally, add $4$ to both sides of the equation to isolate $y$ .
$y=-2x-+4$

STEP 8

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

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QuestionFind the equation of a line through $(-4,4)$ that is perpendicular to $y=\frac{1}{2}x+2$ .

Studdy Solution

STEP 1

Assumptions1. We know the point through which the line passes $(-4,4)$ . We know the line to which our line is perpendicular $y=\frac{1}{} x+$
3. The slope of the 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 $y=mx+b$ is $m$ .So, for the line $y=\frac{1}{2} x+2$ , the slope is $\frac{1}{2}$ .

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 $\frac{1}{2}$ is $-2$ .
So, the slope of the line we are trying to find is $-2$ .

STEP 4

Now that we have the slope of the line, we can use the point-slope form of a line to write the equation of the line. The point-slope form is $y-y1=m(x-x1)$ , where $(x1,y1)$ is a point on the line and $m$ is the slope.

STEP 5

Plug in the values for the slope and the point to write the equation of the line.
$y-4=-2(x+4)$

STEP 6

implify the equation by distributing the $-2$ on the right side of the equation.
$y-4=-2x-8$

STEP 7

Finally, add $4$ to both sides of the equation to isolate $y$ .
$y=-2x-+4$

STEP 8

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

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QuestionFind the equation of a line through (−4,4)(-4,4)(−4,4) that is perpendicular to y=12x+2y=\frac{1}{2}x+2y=21​x+2.

Studdy Solution

STEP 1

Assumptions1. We know the point through which the line passes (−4,4)(-4,4)(−4,4). We know the line to which our line is perpendicular y=1x+y=\frac{1}{} x+y=1​x+3. The slope of the 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+by=mx+b is mmm.So, for the line y=12x+2y=\frac{1}{2} x+2y=21​x+2, the slope is 12\frac{1}{2}21​.

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 12\frac{1}{2}21​ is −2-2−2.So, the slope of the line we are trying to find is −2-2−2.

STEP 4

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

STEP 5

Plug in the values for the slope and the point to write the equation of the line.y−4=−2(x+4)y-4=-2(x+4)y−4=−2(x+4)

STEP 6

implify the equation by distributing the −2-2−2 on the right side of the equation.y−4=−2x−8y-4=-2x-8y−4=−2x−8

STEP 7

Finally, add 444 to both sides of the equation to isolate yyy.y=−2x−+4y=-2x-+4y=−2x−+4

STEP 8

implify the equation to get the final equation of the line.y=−2x−4y=-2x-4y=−2x−4The equation of the line that passes through the point (−4,4)(-4,4)(−4,4) and is perpendicular to the line y=12x+2y=\frac{1}{2} x+2y=21​x+2 is y=−2x−4y=-2x-4y=−2x−4.

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QuestionFind the equation of a line through $(-4,4)$ that is perpendicular to $y=\frac{1}{2}x+2$ .

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

First, we need to find the slope of the given line. The slope of a line in the form $y=mx+b$ is $m$ .So, for the line $y=\frac{1}{2} x+2$ , the slope is $\frac{1}{2}$ .

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 $\frac{1}{2}$ is $-2$ .
So, the slope of the line we are trying to find is $-2$ .

Now that we have the slope of the line, we can use the point-slope form of a line to write the equation of the line. The point-slope form is $y-y1=m(x-x1)$ , where $(x1,y1)$ is a point on the line and $m$ is the slope.

Plug in the values for the slope and the point to write the equation of the line.
$y-4=-2(x+4)$

implify the equation by distributing the $-2$ on the right side of the equation.
$y-4=-2x-8$

Finally, add $4$ to both sides of the equation to isolate $y$ .
$y=-2x-+4$

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