3x + 19y = 7................(1)

x + 2y = 1.....................(2)

We need to determine the intersection points of the lines (1) and (2).

The points of intersection are the points that verifies both equations. Then the points of intersections are the solution of the system.

Let us solve the system.

We will re-write equation (2).

x+ 2y = 1

==> x= 1- 2y

Now we will use the substitution method to solve.

Let us substitute with x= 1- 2y into (1).

3x + 19 y = 7

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

==> 3 - 6y + 19 y = 7

==> 3 + 13y = 7

==> 13y = 4

==> y= 4/13

==> x= 1- 2y = 1- 2*4/13 = 1- 8/13 = 5/13

==> x = 5/13

**Then, the lines intersects at the point ( 5/13, 4/13)**

To find out where the two lines meet, we need to solve the two equations 3x + 19y =7 and x + 2y = 1 for x and y.

3x + 19y =7… (1)

x + 2y = 1… (2)

Now subtracting 3 times equation 2 from equation 1,

(1)- 3*(2)

=> 3x + 19y- 3x - 6y = 7-3

=> 13y = 4

=> y=4/13

Substituting y= 4/13 in x + 2y = 1

x = 1- 2*4/13

=5/13

Therefore x= 5/13 and y= 4/13.

**The lines meet at the point ( 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).**

Where do the lines 3x + 19y =7 and x + 2y = 1 meet.

The lines 3x + 19y =7 and x + 2y = 1 meet at at point (x,y) which satisfy both equations and could be obtained by solving the equations:

3x+19y = 7....(1).

x+2y = 1.......(2).

(1)-3*(2) gives:

(3x+19y) -3(x+2y) = 7-1*3 = 4.

19y-6y = 4.

13y = 4.

y = 4/19.

Put y = 4/1 in the equation (2),x+2y = 1:

x+2(4/19) = 1.

x = 1-8/19 = 19/19-7/19 = 12/19.

Therefore x = 12/19 and y = 7/19.