SOLUTION: . Find the equation of the line passing through the point (3, 7) and perpendicular to the line 3y = 4 – 2x

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: . Find the equation of the line passing through the point (3, 7) and perpendicular to the line 3y = 4 – 2x      Log On


   

Question 1102492: . Find the equation of the line passing through the point (3, 7) and
perpendicular to the line 3y = 4 – 2x
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52802) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find the equation of the line passing through the point (3, 7) and
perpendicular to the line 3y = 4 – 2x
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

The given line  y = %28-2%2F3%29%2Ax + 4%2F3 has the slope of -2%2F3.

Since the projected/requested line is perpendicular to the given line, it has the slope value opposite to reciprocal, i.e. 3%2F2.

Hence, the projected/requested line has an equation of the form  y = %283%2F2%29%2Ax+%2B+b  with unknown coefficient "b".

To find "b", simply substitute the coordinates of the given point x= 3 and y= 7 respectively into this equation  y = %283%2F2%29%2Ax+%2B+b.  You will get

7 = %283%2F2%29%2A3+%2B+b,   or  14 = 3*3 + 2b,

which implies  2b = 14 - 9 = 5  and  b = 5%2F2.


Thus your final equation of the projected/requested straight line is 

y = %283%2F2%29%2Ax+%2B+5%2F2,  or  2y = 3x + 5.


Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Here is another way -- that I find easier and faster -- to answer questions like this about lines parallel or perpendicular to a given line.

(1) Put the equation in the form Ax+By=C.

(2a) Every line parallel to the given line will have an equation also of the form Ax+By=K for some constant K. (If K is the same constant as C, the lines are of course the same line, not parallel lines.)

(2b) Every line perpendicular to the given line will have an equation of the form Bx-Ay=K for some constant K. Note that the two coefficients have switched places, and one of them has changed sign.

Let's apply this method to your problem, where we want an equation of the line through (3,7) perpendicular to the equation 3y = 4-2x.

Step 1: put the equation in the required form: 2x+3y = 4.

Step 2: Since we want a perpendicular line, it will have an equation of the form 3x-2y = K, where K is some constant.

The constant is easily determined, knowing that the coordinates of the given point must satisfy the equation:
3%283%29-2%287%29+=+9-14+=+-5

The equation (in this form) of the line we are looking for is 3x-2y = -5.