QuestionFind the equations of lines through $(-2,3)$ : one parallel to $y=-2x+4$ and one perpendicular to it.

Studdy Solution

STEP 1

Assumptions1. The point through which the lines pass is $(-,3)$ . The given line has the equation $y=-x+4$
3. The slope of a line parallel to a given line is the same as the slope of the given line4. 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 $y=mx+b$ is $m$ .
In the given equation $y=-2x+4$ , the slope is $-2$ .

STEP 3

Now, we can find the equation of the line that is parallel to the given line. The slope of this line is the same as the slope of the given line, which is $-2$ .
The equation of a line in slope-intercept form is $y=mx+b$ , where $m$ is the slope and $b$ is the y-intercept. We can find $b$ by substituting the given point $(-2,3)$ into the equation and solving for $b$ .
$3 = -2(-2) + b$

STEP 4

olve the equation for $b$ .
$3 =4 + b$ $b =3 -4$ $b = -1$

STEP 5

Now we can write the equation of the line that is parallel to the given line and passes through the point $(-2,3)$ .
$y = -2x -1$

STEP 6

Next, we can find the equation of the line that is perpendicular to the given line. The slope of this line is the negative reciprocal of the slope of the given line. The negative reciprocal of $-2$ is $1/2$ .
Again, we can find $b$ by substituting the given point $(-2,3)$ into the equation and solving for $b$ .
$3 = \frac{1}{2}(-2) + b$

STEP 7

olve the equation for $b$ .
$3 = -1 + b$ $b =3 +1$ $b =4$

STEP 8

Now we can write the equation of the line that is perpendicular to the given line and passes through the point $(-2,3)$ .
$y = \frac{1}{2}x +4$ So, the equation of the line that is parallel to the given line and passes through the point $(-2,3)$ is $y = -2x -1$ , and the equation of the line that is perpendicular to the given line and passes through the point $(-2,3)$ is $y = \frac{1}{2}x +4$ .

Was this helpful?

QuestionFind the equations of lines through $(-2,3)$ : one parallel to $y=-2x+4$ and one perpendicular to it.

Studdy Solution

STEP 1

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 $m$ .
In the given equation $y=-2x+4$ , the slope is $-2$ .

STEP 3

STEP 4

olve the equation for $b$ .
$3 =4 + b$ $b =3 -4$ $b = -1$

STEP 5

Now we can write the equation of the line that is parallel to the given line and passes through the point $(-2,3)$ .
$y = -2x -1$

STEP 6

STEP 7

olve the equation for $b$ .
$3 = -1 + b$ $b =3 +1$ $b =4$

STEP 8

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionFind the equations of lines through (−2,3)(-2,3)(−2,3): one parallel to y=−2x+4y=-2x+4y=−2x+4 and one perpendicular to it.

Studdy Solution

STEP 1

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.In the given equation y=−2x+4y=-2x+4y=−2x+4, the slope is −2-2−2.

STEP 3

STEP 4

olve the equation for bbb.3=4+b3 =4 + b3=4+bb=3−4b =3 -4b=3−4b=−1b = -1b=−1

STEP 5

Now we can write the equation of the line that is parallel to the given line and passes through the point (−2,3)(-2,3)(−2,3).y=−2x−1y = -2x -1y=−2x−1

STEP 6

STEP 7

olve the equation for bbb.3=−1+b3 = -1 + b3=−1+bb=3+1b =3 +1b=3+1b=4b =4b=4

STEP 8

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionFind the equations of lines through $(-2,3)$ : one parallel to $y=-2x+4$ and one perpendicular to it.

First, we need to find the slope of the given line. The slope of a line in the form $y=mx+b$ is $m$ .
In the given equation $y=-2x+4$ , the slope is $-2$ .

olve the equation for $b$ .
$3 =4 + b$ $b =3 -4$ $b = -1$

Now we can write the equation of the line that is parallel to the given line and passes through the point $(-2,3)$ .
$y = -2x -1$

olve the equation for $b$ .
$3 = -1 + b$ $b =3 +1$ $b =4$