The equations 3x + 19y =7 and x + 2y = 1 have to solved simultaneously to arrive at the point where they meet.

x + 2y = 1

=> x = 1- 2y

substitute in 3x + 19y = 7

=> 3 - 6y + 19y = 7

=> 13y = 4

=> y = 4/13

x = 5/13

**The point of contact is (5/13 , 4/13)**

The x and y coordinates of the lines 3x + 19y =7 and x + 2y = 1 are the same at the point where they intersect. To determine the point solve the system of equations.

x + 2y = 1 gives x = 1 - 2y

Substitute this for x in 3x + 19y = 7

3(1 - 2y) + 19y = 7

3 - 6y + 19y = 7

13y = 4

y = 4/13

x = 1 - 8/13 = 5/13

The point of intersection of the two lines is `(5/13, 4/13)`

The intercepting point of the lines is the solution of the system formed from the equations of the lines.

We'll solve the system using matrix.

We'll use determinant of the system to check if the system has solution or not.

3x + 19y =7 (1)

x + 2y = 1 (2)

det A = 3*2 - 19*1

det A = 6 - 19

det A = -13

Since the determinant is not cancelling, the system has a unique solution, namely the intercepting point of the given lines.

x = det x/det A

det x = 7*2 - 19

det x = 14-19

det x = -5

x = -5/-13

**x = 5/13**

y = det y/det A

det y = 3*1 - 1*7

det y = -4

y = -4/-13

**y = 4/13**

**The intercepting point of the given lines is the solution of the system: (5/13 ; 4/13).**