Question

Which of the following is the equation of a line perpendicular to the line
`y = (-3)/(2) x + 4`, passing through the point (3,9)?

A. 2x - 3y = 21
B. -2x +3y = 21
C. -2x - 3y = -21
D. 2x +3y = 21

Answer 1

Choice B is correct answer.

Explanation:

Two lines are perpendicular if their slopes negative reciprocals to each other.

y = mx+c is equation of line where m is slope and c is y-intercept.

Given line is

y = -3/2x+4 where slope is -3/2.

given point is:

(x,y) = (3,9)

We have to find the equation of line which is perpendicular to given line.

hence, slope of line that is perpendicular to y = -3/2x+4 is 2/3.

Equation of perpendicular line is:

y = 2/3x+c

We have to find y-intercept:

putting given point in above equation we have,

9 = 2/3(3)+c

9 = 2 +c

c= 7

hence , put the value of y-intercept in above equation, we have

y = 2/3x+7

y = 2x+21/3

3y = 2x+21

-2x+3y = 21 is equation of line perpendicular to y = -3/2x+4.

Answer 2

For this case, we have that by definition, if two lines are perpendicular, it follows that:

`m_ {1} * m_ {2} = - 1`

If we have the line:
`y = - \frac {3} {2} x + 4`

With slope
`m_ {1} = - \frac {3} {2}`

A line perpendicular to this would have slope:

`m_ {2} = \frac {-1} {- \frac {3} {2}}\\m_ {2} = \frac {2} {3}`

Thus, the equation of this line is given by:

`y = \frac {2} {3} x + b`

Substituting the point
`(x, y) = (3,9)`we find the cut point "b":

`9 = \frac {2} {3} 3 + b\\9 = 2 + b\\b = 9-2\\b = 7`

Thus, the equation is:

`y = \frac {2} {3} x + 7\\y- \frac {2} {3} x = 7`

Multiplying by "3" on both sides of the equation:

`3y-2x = 21`

Answer:

Option B