Line l passes through point (12,2) and line p is the graph of 4x-7y=22 . If l⊥p , what is the equation of l ?

Question

Line l passes through point (12,2) and line p is the graph of 4x-7y=22 .

If l⊥p , what is the equation of l ?

Answer 1

The slope of line p is 4/7. Any line perpendicular to it must have a slope of -7/4. Using the point-slope form of a linear equation (m = (y - y₀)/(x - x₀)) gives

-7/4 = (y - 2)/(x - 12)

Cross-multiplying, (-7/4)(x - 12) = (y - 2)

Simplifying into standard form,

-7(x - 12) = 4(y - 2)

-7x + 84 = 4y - 8

-7x -4y = -92

7x + 4y = 92

Answer 2

The problem statement tells you that line l and line p are perpendicular. This information tells us that the slopes of l and p will be negative reciprocals of eachother. Reminder: the reciprocal of something is found by flipping the numerator and denominator of a fraction (example: reciprocal of 3/7 is 7/3).

First, solve the equation for line p to isolate y on one side.

4x-7y = 22

-7y = 22-4x

7y = 4x-22

y = (4/7)x - 22/7

This shows us that the slope of l must be -1*(7/4) = -7/4. Take this slope and plug the slope and given point on l into slope intercept form (y=mx+b where m is the slope and b is the y-intercept).

2=(-7/4)*(12) +b

2 = -21 + b

b= 23

So the equation for l is y = (-7/4)x+23

Answer 3

Hadassah

4x - 7y = 22

4x = 22 + 7y

(4/7)x = (22/7) + y

y = (4/7)x - (22/7)

So the slope (gradient) of line p is 4/7

Line l is perpendicular to p. so the product of their gradients must be -1

That means the gradient of line l must be -7/4

So for line l

y=mx+c

m = -7/4

when x=12 y = 2

so 2 = (-7/4).12 + c

2 = -21 + c

c = 23

so y = (-7/4) x + 23 is the equation of line l

Mike