Solved on Dec 15, 2023

Find the equation of a line passing through $(-2,2)$ and perpendicular to $y=\frac{1}{2} x-3$ . A. $y=-2 x+2$ B. $y=-2 x-2$ C. $y=\frac{1}{2} x-2$ D. $y=\frac{1}{2} x+2$

STEP 1

Assumptions
1. The given line has the equation $y=\frac{1}{2}x-3$ .
2. The slope of the given line is $\frac{1}{2}$ .
3. A line perpendicular to the given line will have a slope that is the negative reciprocal of $\frac{1}{2}$ .
4. The line we are looking for passes through the point $(-2,2)$ .
5. The equation of the line we are looking for will be in slope-intercept form, which is $y=mx+b$ , where $m$ is the slope and $b$ is the y-intercept.

STEP 2

Calculate the slope of the line perpendicular to the given line by taking the negative reciprocal of the slope of the given line.
$m_{perpendicular} = -\frac{1}{m_{given}}$

STEP 3

Substitute the given slope into the formula to find the slope of the perpendicular line.
$m_{perpendicular} = -\frac{1}{\frac{1}{2}}$

STEP 4

Calculate the slope of the perpendicular line.
$m_{perpendicular} = -2$

STEP 5

Now that we have the slope of the perpendicular line, we can use the point-slope form of a line to find the equation of the line that passes through the point $(-2,2)$ . The point-slope form is given by:
$y - y_1 = m(x - x_1)$
where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

STEP 6

Substitute the slope of the perpendicular line and the coordinates of the given point into the point-slope form equation.
$y - 2 = -2(x - (-2))$

STEP 7

Simplify the equation by distributing the slope and moving the point coordinates.
$y - 2 = -2(x + 2)$

STEP 8

Distribute the slope $-2$ across the terms in the parentheses.
$y - 2 = -2x - 4$

STEP 9

Add 2 to both sides of the equation to solve for $y$ and put it into slope-intercept form.
$y = -2x - 4 + 2$

STEP 10

Combine like terms to get the final equation of the line.
$y = -2x - 2$
The equation of the line in slope-intercept form that passes through $(-2,2)$ and is perpendicular to the graph of $y=\frac{1}{2}x-3$ is $y=-2x-2$ .
Therefore, the correct answer is B. $y=-2x-2$ .

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Find the equation of a line passing through $(-2,2)$ and perpendicular to $y=\frac{1}{2} x-3$ . A. $y=-2 x+2$ B. $y=-2 x-2$ C. $y=\frac{1}{2} x-2$ D. $y=\frac{1}{2} x+2$

STEP 1

STEP 2

Calculate the slope of the line perpendicular to the given line by taking the negative reciprocal of the slope of the given line.
$m_{perpendicular} = -\frac{1}{m_{given}}$

STEP 3

Substitute the given slope into the formula to find the slope of the perpendicular line.
$m_{perpendicular} = -\frac{1}{\frac{1}{2}}$

STEP 4

Calculate the slope of the perpendicular line.
$m_{perpendicular} = -2$

STEP 5

Now that we have the slope of the perpendicular line, we can use the point-slope form of a line to find the equation of the line that passes through the point $(-2,2)$ . The point-slope form is given by:
$y - y_1 = m(x - x_1)$
where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

STEP 6

Substitute the slope of the perpendicular line and the coordinates of the given point into the point-slope form equation.
$y - 2 = -2(x - (-2))$

STEP 7

Simplify the equation by distributing the slope and moving the point coordinates.
$y - 2 = -2(x + 2)$

STEP 8

Distribute the slope $-2$ across the terms in the parentheses.
$y - 2 = -2x - 4$

STEP 9

Add 2 to both sides of the equation to solve for $y$ and put it into slope-intercept form.
$y = -2x - 4 + 2$

STEP 10

Combine like terms to get the final equation of the line.
$y = -2x - 2$
The equation of the line in slope-intercept form that passes through $(-2,2)$ and is perpendicular to the graph of $y=\frac{1}{2}x-3$ is $y=-2x-2$ .
Therefore, the correct answer is B. $y=-2x-2$ .

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Find the equation of a line passing through (−2,2)(-2,2)(−2,2) and perpendicular to y=12x−3y=\frac{1}{2} x-3y=21​x−3. A. y=−2x+2y=-2 x+2y=−2x+2 B. y=−2x−2y=-2 x-2y=−2x−2 C. y=12x−2y=\frac{1}{2} x-2y=21​x−2 D. y=12x+2y=\frac{1}{2} x+2y=21​x+2

STEP 1

STEP 2

Calculate the slope of the line perpendicular to the given line by taking the negative reciprocal of the slope of the given line.mperpendicular=−1mgivenm_{perpendicular} = -\frac{1}{m_{given}}mperpendicular​=−mgiven​1​

STEP 3

Substitute the given slope into the formula to find the slope of the perpendicular line.mperpendicular=−112m_{perpendicular} = -\frac{1}{\frac{1}{2}}mperpendicular​=−21​1​

STEP 4

Calculate the slope of the perpendicular line.mperpendicular=−2m_{perpendicular} = -2mperpendicular​=−2

STEP 5

STEP 6

Substitute the slope of the perpendicular line and the coordinates of the given point into the point-slope form equation.y−2=−2(x−(−2))y - 2 = -2(x - (-2))y−2=−2(x−(−2))

STEP 7

Simplify the equation by distributing the slope and moving the point coordinates.y−2=−2(x+2)y - 2 = -2(x + 2)y−2=−2(x+2)

STEP 8

Distribute the slope −2-2−2 across the terms in the parentheses.y−2=−2x−4y - 2 = -2x - 4y−2=−2x−4

STEP 9

Add 2 to both sides of the equation to solve for yyy and put it into slope-intercept form.y=−2x−4+2y = -2x - 4 + 2y=−2x−4+2

STEP 10

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Find the equation of a line passing through $(-2,2)$ and perpendicular to $y=\frac{1}{2} x-3$ . A. $y=-2 x+2$ B. $y=-2 x-2$ C. $y=\frac{1}{2} x-2$ D. $y=\frac{1}{2} x+2$

Calculate the slope of the line perpendicular to the given line by taking the negative reciprocal of the slope of the given line.
$m_{perpendicular} = -\frac{1}{m_{given}}$

Substitute the given slope into the formula to find the slope of the perpendicular line.
$m_{perpendicular} = -\frac{1}{\frac{1}{2}}$

Calculate the slope of the perpendicular line.
$m_{perpendicular} = -2$

Substitute the slope of the perpendicular line and the coordinates of the given point into the point-slope form equation.
$y - 2 = -2(x - (-2))$

Simplify the equation by distributing the slope and moving the point coordinates.
$y - 2 = -2(x + 2)$

Distribute the slope $-2$ across the terms in the parentheses.
$y - 2 = -2x - 4$

Add 2 to both sides of the equation to solve for $y$ and put it into slope-intercept form.
$y = -2x - 4 + 2$