Verify if the lines have common points 2x - 5y +2 = 0 and y = -3x + 6 ?

The lines given are 2x - 5y +2 = 0  and y = -3x +
6

At the point of intersection of two lines, the
x-coordinate as well as the y-coordinate is the same.

2x -
5y +2 = 0

=> x = (-2 +
5y)/2

substitute in y = -3x +
6

=> y = -3*(-2 + 5y)/2 +
6

=> y = 6/2 - 15y/2 +
6

=> y = 9 -
15y/2

=> y + 15y/2 =
9

=> 17y/2 =
9

=> y = 18/17

x = (-2
+ 5y)/2 = (-2 + 5*(18/17))/2 = (-34 + 90)/34

=> x =
56/34 = 28/17

This gives the common point of
the lines as (28/17, 18/17)