Use a matrix approach to solve the system

Question

x − 3y − z = 13
3x + y − 4z = −13
−2x + 5y + 3z = −19

 

x = ?

y = ?

z = ?

 

(If the system has infinitely many solutions, express your answer in terms of k, where x = x(k), y = y(k), and z = k. If the system is inconsistent, enter INCONSISTENT.)

 

I already got y = -5 and z = 2 but am not getting a correct answer for x. I would be very grateful for some assistance!

 

 

Answer 1

Set up your matrix.

 

| 1 -3 -1|  13|

| 3  1 -4| -13|

|-2  5  3| -19|

 

Initiate Gaussian Reduction

R2=R2-3R1

R3=R3+2R1

 

| 1 -3  -1| 13|
| 0 10 -1| -52|
| 0 -1   1|  7  |

 

R3=-R3

Switch R2 and R3

 

| 1 -3 -1 | 13|
| 0  1 -1 | -7 |

| 0 10 -1| -52|

 

R3=R3-10R2

 

| 1 -3 -1 | 13|
| 0  1 -1 | -7 |
| 0  0  9 | 18 |

 

R3=(1/9)R3

 

| 1 -3 -1 | 13|
| 0 1  -1 | -7 |
| 0 0   1 |  2 |

 

R2=R2+R3

R1=R1+R3

 

| 1 -3 0  | 15|
| 0  1  0 | -5 |
| 0  0  1 | 2  |

 

R1=R1+3R2

 

| 1 0 0 |  0 |
| 0 1 0 | -5 |
| 0 0 1 |  2 |

 

x=0

y=-5

z=2

Answer 2

Dear Katelyn,

 

I have a solution for you:

 

 

x = 0

y = -5

z = 2

 

I did not do this using matrices, rather a TI-89 Titanium calculator.  

 

Personally, Katelyn, I never learned how to use matrices.  I don't find them any easier to use that brute force algebra.