Question

find the equation of the line through the intersection of 5 x minus 3 Y equal to 1 and 2 X + 3 Y - 23 equal to zero and perpendicular to the line 5 x minus 3 Y - 1 equal to zero​

Answer 1

Answer:

3x + 5y = 781 / 21

Step-by-step explanation:

Three steps:

1. Get the point P that is the intersection of the two given lines
2. Get a and b in the equation ax + by = c for the required line.  This is essentially getting the slope.
3. Get c in the equation ax + by = c so that the line passes through P.

Step 1

Let P=(x,y).  Then...

5x - 3y = 1      ... (1)

2x + 3y = 23      ... (2)

Adding these two equations gives

7x = 24  =>  x = 24 / 7,

so 3y = 5x - 1 = 120/7  - 1 = 113 / 7    =>   y = 113 / 21

Thus P = ( 24 / 7,  113 / 21 ).

Step 2

The line 5x - 3y = 1 has slope 5/3.

The slope of lines perpendicular to this is -3/5.

So the line we seek has the form

3x + 5y = c.

Step 3

Putting the coordinates of P into the equation 3x + 5y = c for the line gives:

c = 3x + 5y

  = 3 ( 24 / 7 )  +  5 ( 113 / 21 )

  = 72 / 7  +  565 / 21

  = 216 / 21  +  565 / 21

  = 781 / 21.

Therefore the equation of the line is:

3x + 5y = 781 / 21