Solve :

5x - 7y = 12

5y - 2x = - 7

There is one solution:

use elimination method

5x - 7y = 12
5y - 2x = -7    ---> change equation around

 5x - 7y = 12
-2x + 5y = -7

2 x ( 5x - 7y) = 2 x (12)    multiply both sides by 2
5 x (-2x +5y) = 5 x (-7)

this give you
 10x - 14y =24 
-10x+25y = -35                    add down

        11y = - 11                    x is eliminated to find y value
           
            y = -1                         input to one of the original equations

5(-1)  - 2x = -7

-5  - 2x = -7
+5           +5                       add 5 to both sides

-2x = -2
    x = 1

your coordinates for when they intersect is at (1, -1)
one solution

5x - 7y = 12
5y - 2x = -7
5x - 7y = 12
-2x + 5y = -7
3x + 2y = 5
3(2) + 2(-1) = 5
6 + (-1) = 5
6 - 1 = 5
5 = 5