Verified answer

Okay, so you are given the point (4, -4).

And you know that the line is perpendicular to 5y = x - 10 (which means it's slope will be the opposite reciprocal, by definition of perpendicular lines).

So first begin by solving 5y = x - 10

Divide both sides by 5, leaving you with:

y = 1/5x - 2

If the line whose equation you are looking for is perpendicular to this line, you know that it's slope is going to be the opposite reciprocal (or negative inverse) of 1/5, which is -5/1 = -5.

Going back to the point you were given, (4, -4), you can easily solve now that you know the slope is going to be -5. So simply use point-slope form (y-y1 = m(x-x1))

Like this:

y - (-4) = -5(x - 4)

y + 4 = -5x + 20

y = -5x + 16

And there you have your equation. In function notation, this could be written as f(x) = -5x + 16

2. You now are looking for the equation of a line which is parallel to the line of 6x + 5y = 11. Because the line is parallel, then, by definition of parallel lines you will have the same slope. So let's solve for 6x + 5y = 11 to get a better look:

6x + 5y = 11

5y = -6x + 11

y = -6/5x +11/5

Because the slope is -6/5x, the line you are looking for also has a slope of -6/5x, again, because they are parallel.

So with the given point (-5, -8), and now knowing the slope is -6/5x, you can easily find the equation of the line using point-slope.

Like this:

y - (-8) = -6/5(x - (-5))

y + 8 = -6/5x + 6 (because 30/5 simplifies to 6)

y = -6/5x - 2

And there you have the equation, which can be written as f(x) = -6/5x - 2 in function notation.

Hope this helps!

1) Perpendicular lines have slopes that are negative reciprocals. That is the multiply to give -1. To find the solve, convert to slope-intercept form.

5y = x - 10

y = (1/5)x - 2

So the slope of this line is 1/5 = 0.2 . The negative reciprocal is -5. So the slope of the target line is -5.

Using point-slope form: y-y1 = m(x-x1), where m is the slope and (x1, y1) is a point on the line.

y - (-4) = -5(x - 4)

y + 4 = -5(x-4)

Expand and simplify to get to slope-intercept form.

y = -5(x-4) - 4

y = -5x + 20 - 4

y = -5x + 16

Convert to function notation.

f(x) = -5x + 16

2) Parallel lines have the same slope. So determine the slope of the reference line by converting to slope-intercept form.

6x + 5y = 11

5y = -6x + 11

y = (-6/5)x + 11/5

So the slope of the reference line is -6/5. The negative reciprocal is 5/6. Using point-slope form to get the equation.

y-(-8) = (5/6)(x - -5)

y + 8 = (5/6)(x + 5)

Convert to slope-intercept form.

y = (5/6)x + 25/6 - 8

y = (5/6)x + 25/6 - 48/6

y = (5/6)x - 23/6

Convert to function notation.

f(x) = (5/6)x - 23/6