Question 62306
You can solve this as a matrix, but you can also elimination and substitution:
:
x - 2y - 3z = 4
2x -4y + 5z = -3
5x -6y + 4z = -7
:
Let's just deal with the 1st & 2nd equation:
x - 2y - 3z = 4
2x -4y + 5z = -3
:
Mult the 1st equation by 2, leave the 2nd equation as it is:
2x - 4y - 6z = 8
2x - 4y + 5z = -3
---------------------subtract & we eliminate x & y, it's easy to find z!
0x + 0y -11z = 11
z = 11/-11
z = -1
:
Now that we know z = -1 we can rewrite the 3 equations:
x - 2y - 3(-1) = 4  >>  x -2y + 3 = 4  >> x - 2y = 4 - 3
2x -4y + 5(-1) = -3 >> 2x -4y - 5 = -3 >> 2x -4y = -3 + 5
5x -6y + 4(-1) = -7 >> 5x -6y - 4 = -7 >> 5x - 6y = -7 + 4
We end up with 3 equation with 2 unknowns:
x - 2y = +1
2x -4y = +2
5x -6y = -3
:
Use the 1st equation for substitution:
x = 2y + 1
:
Substitute for x in the 3rd "2 unknown" equation:
5(2y+1) - 6y = -3
10y + 5 - 6y = -3
4y = -3 - 5
4y = -8
y = -8/4
y = -2
:
Find x using x = 2y +1
x = 2(-2) + 1
x = -4 + 1
x = -3
:
We have x = -3, y = -2; z = -1
Choose one of the original equations to check our solutions, the 2nd equation:
2x -4y + 5z = -3
2(-3) - 4(-2) + 5(-1) = -3
-6 + 8 - 5 = -3; equality reigns
:
:
:
and 
2x - y = -1
-2x + z = 1
y - z = 0
:
Look at the equation y - z = 0, that means y = z, right, 
Let's replace y with z in the 1st equation and pair it with the 2nd equation:
2x  - z = -1
-2x + z = 1 
------------  add these and you get
0  +  0 =  0
:
That means there is no unique solution to this system