What is the solution of the system of equations?

When we are given a system of 3 linear equations, with 3 variables, we proceed as follows:

We consider 2 pairs or equations, for example (1, 2) and (2, 3), and eliminate one of the variables in each pair, creating a system of 2 linear equations with 2 unknowns. 

Note that the third equation contains -2y which can be used to eliminate easily -6y in the second equation, and -4y in the fourth.

i) consider equations 1 and 3:

-3x-4y-3z=-7
5x-2y+5z=9

multiply the second equation by -2:

-3x-4y-3z=-7
-10x+4y-10z=-18

adding the 2 equations we have -13x-13z=-25

ii) consider equations 2 and 3. Multiply the third equation by -3:

    2x-6y+2z=3
-15x+6y-15z=-27

adding the 2 equations we have -13x-13z=-24

So we got -13x-13z is -25, but also -24. this means the system is inconsistent, so it has no solution.

the system has no solutions