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# Use a matrix approach to solve the system

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!

### 2 Answers by Expert Tutors

Carlos T. | Math Tutoring From Basic Algebra to Advanced CalculusMath Tutoring From Basic Algebra to Adva...
5.0 5.0 (60 lesson ratings) (60)
1

| 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

Thank you so much Carlos!
Welcome! Best of luck.
William S. | Experienced scientist, mathematician and instructor - WilliamExperienced scientist, mathematician and...
4.4 4.4 (10 lesson ratings) (10)
0
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.