augmented matrix

Question

solve the following system by row-reducing the augmented matrix.  Show all solutions if there be any.

 

x + y + z = 1

3x - y - z = 4

x + 5y + 5z = -1

Answer 1

|1  1  1  1|

|3 -1 -1  4|

|1  5  5 -1|

 

Subtract 3R1 from R2.  Subtract R1 from R3:

|1  1  1  1|

|0 -4 -4  1|

|0  4  4 -2|

 

Multiply R1 by 4:

|4  4  4 4|

|0 -4 -4 1|

|0  4  4 -2|

 

Add R1 and R2, and place the result in R1.  Add R3 and R2, and place the result in R3:

|4  0  0 5|

|0 -4 -4 1|

|0  0  0 -1|

 

That last row of all 0s for your variable coefficients, coupled with a -1 for your constant, lets you know it's an inconsistent system.  No solution.