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

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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)**

The given lines are intercepting if the system of equations of the lines has a solution.

We'll solve the system of equations of the lines using substitution method:

y = -3x+6 (1)

2x-5y+2=0 (2)

2x - 5(-3x+6) + 2 = 0

We'll remove the brackets:

2x + 15x - 30 + 2 = 0

17x = 28

x = 28/17

y = -3x+6 => y = (-3*28 + 102)/17

y = 18/17

**The lines are intercepting and the coordinates of the intercepting point are: (28/17 ; 18/17).**