Find the linear equation that is through the point (-7, 5) and perpendicular to the linear equation 3x-8y-11.?

Question

Express the equation in the standard form and the slope intercept form. The answers are 8x+3y+41 =0 and y=(-8/3)x - (41/3) but I don't know how to work out the problem.

Answer 1

Perpendicular to the other line means that the slope of this line is the negative reciprocal of the given one.

First, put the other line in slope-intercept form:

3x - 8y - 11 = 0
8y = 3x - 11
y = (3/8)x - 11

Its slope is 3/8, so the slope of the line that you must construct is -1/(3/8) = -8/3.

y = -8/3 * x + b

Now you can plug in (-7,5) for x and y to compute b:

y = -8/3 * x + b
5 = -8/3 * (-7) + b
5 = 56/3 + b
b = 5 - 56/3
b = 15/3 - 56/3
b = -41/3

That gives us the slope-intercept equation of the line:

y = -8/3 * x + (-41/3)
y = -8/3 x - 41/3

To place in standard form, you'd multiply by 3 to eliminate the denominators and move everything to the same side as "y":

y = -8/3 x - 41/3
3y = -8x - 41
8x + 3y + 41 = 0

Answer 2

Because the line you are looking for is perpendicular to 3x-8y=11, its slope will be the negative reciprocal of 3x-8y=11.
Let's put 3x-8y=11 into standard form y=mx+b

3x-8y=11
-8y =11-3x
-8y = -3x +11
8y =3x -11
y = 3/8x -11/8

The slope m is 3/8

Now on to the solution
The line you want has the slope -8/3(the negative reciprocal).
WE have, therefore, y = -8/3x +b

If only we knew what the value of b was!
We can find it. Since (-7,5) is a point on the line, it satisfies the equation y = -8/3x + b
Substituting, we get 5 = -8/3(-7) + b
5 = 56/3 + b
15/3 - 56/3 = b
-41/3 = b
We're done. The equation you seek is y=-8/3x-41/3
We can tidy this up. Multiply both sides by 3 to get
3y=-8x-41, or 8x+3y+41=0

Answer 3

y = mx + b

If the lines are parallel, then the slope, m, is the same.

5 = (-7 * 3/8) + b
b = 5 + 21/8 = 7.625

Line equation is y = .375 x + 7.625
8y = 3x + 61

Answer 4

Short way:
The equation has the form
8x+3y-C = 0
Plug in (-7,5)
8(-7)+3(5)-C = 0
C = 41
Therefore, the equation is
8x+3y+41 = 0
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Reason:
Ax+Bx+c = 0
is perpendicular to
Bx-Ay+d = 0