# Write the equation of a line that passes through (-4,2) and is perpendicular to 3x+6y=9

## First of all they should have reduced your given 3x+6y=9

to x + 2y = 3

quickest way:
since the new equation is perpendicular to the old one, it
should be
2x - y = c , (realizing that for ax + by = c, the slope is -a/b )

-8 - 2 = c
c =-10

new equation: 2x - y = -10

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## To find the equation of a line that is perpendicular to another line, we first need to determine the slope of the given line.

The given line has the equation 3x + 6y = 9 in standard form. To find the slope, we need to rewrite this equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

Starting with 3x + 6y = 9, let's isolate y by subtracting 3x from both sides:

6y = -3x + 9

Next, divide both sides by 6 to solve for y:

y = (-1/2)x + 3/2

Now that we have the slope-intercept form, we can observe that the slope of the given line is -1/2.

Since we are looking for a line that is perpendicular to this line, we can determine the new line's slope by taking the negative reciprocal of -1/2. The negative reciprocal of a number is obtained by flipping its sign and then taking the reciprocal.

Therefore, the slope of the new line is 2.

Now, using the point-slope form of a linear equation, we can plug in the given point (-4, 2) and the slope (2) to find the equation of the perpendicular line:

y - y1 = m(x - x1)

Substituting the values (-4, 2) and m = 2:

y - 2 = 2(x - (-4))

y - 2 = 2(x + 4)

Simplifying further:

y - 2 = 2x + 8

Finally, rearrange the equation to the slope-intercept form:

y = 2x + 8 + 2

y = 2x + 10

Therefore, the equation of the line that passes through (-4, 2) and is perpendicular to 3x + 6y = 9 is y = 2x + 10.

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## To find the equation of a line that is perpendicular to another line, we need to determine the slope of the given line and then find the negative reciprocal of that slope.

Given that the equation of the given line is 3x + 6y = 9, we can rewrite it in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.

To convert the given equation to slope-intercept form, let's solve it for y:
6y = -3x + 9
y = (-3/6)x + 9/6
Simplifying, we get:
y = (-1/2)x + 3/2

So, the slope of the given line is -1/2.

To find the slope of the line perpendicular to this, we take the negative reciprocal of the slope:
Perpendicular slope = -1 / (-1/2)
Perpendicular slope = 2

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

Given that the line passes through (-4, 2) and has a slope of 2, we can substitute these values into the point-slope form:
y - 2 = 2(x - (-4))

Simplifying, we get:
y - 2 = 2(x + 4)
y - 2 = 2x + 8
y = 2x + 10

Therefore, the equation of the line that passes through (-4,2) and is perpendicular to 3x + 6y = 9 is y = 2x + 10.