a) Solution by Cremer's rule
And the solution is:
b) Solution by row transformation
The augmented matrix will be: 

Description 
Operation 
Result 
Now multiply each value in the second row by −2 and add it to the first row. 


Multiply each value in the 1st row by 3 and 3rd row by −2 and add. 


Multiply 3rd row by 5 and add each item with the second row. 


The last result matrix is equivalent to the equations: 

Now it is clear that 64z = 192 and z = 192/64 = 3
For the second row we have: −5y + 4z = 17 and y = (17 − 4z)/−5 = (17 − 4 · 3)/−5 = −1
From the first row we have 2x − y + 2z = 11
and x = (11 + y − 2z)/2 = (11 − 1 − 2 · 3)/2 = 2
And we get the solution as before: x = 2 y = −1 z = 3
c) Solution by elimination
From the first equation we find the value of y:
From the first equation we find the value of y: 
y = 2x + 2z − 11 
(1) 
Substitute value of y to the 2nd and 3rd equation: 
x + 2(2x + 2z − 11) − z = −1 


5x + 3z = 19 
(2) 

3x − 2(2x + 2z  11) − 3z = −1 


x = 23 − 7z 
(3) 
Substitute equation (3) into eq. (2) 
5(23 − 7z) + 3z = 19 


32z = 96 


z = 3 

From eq. (3) we have 
x = 23 − 7z = 23 − 21 = 2 

From eq. (1) we get 
y = 2x + 2z − 11 = −1 

And again, the solution is as before: x = 2 y = −1 z = 3
