Do the lines x+ 3y +7 =0, 3x+2y +11=0 and 2x+y+4=0 have a common point of interception?

To determine if the 3 lines have a common point, we'll
have to determine the solution of the system formed by the equations of the 3
lines:

x+ 3y +7 =0

We'll
subtract 7 both sides:

x + 3y = -7
(1)

3x+2y +11=0

We'll
subtract 11 both sides:

3x + 2y = -11
(2)

2x+y+4=0

We'll subtract 4
both sides:

2x + y = -4
(3)

We'll determine the matrix of the system. The
determinant is formed from the coefficients of x and
y.

           1    3

    A
=  3    2

           
2    1

We'll take the first2 lines and columns of the
matrix and we'll calculate the minor of the matrix
A.

         1    3

d
=   

         3     2 

d = 1*2
- 3*3 = 2 - 9 = -7

Now, we'll calculate the determinant
that we'll tell us if the system has solution or not.

This
determinant will be formed from the minor, the last row and the column of the terms from
the right side of equal:

        1    3   
-7

C =  3     2   -11

       
2    1    -4

C = -2*4 - 7*3 - 3*2*11 + 2*2*7 + 11 +
3*3*4

C = - 8 - 21 - 66 + 28 + 11 +
36

C = 20  - 1 - 30

C =
-11

Since C is not zero, the system has no
solutions.

If the system doesn't have
solutions, that means that the lines do not intercepting or do not have any common
point.