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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!
 
 
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2 Answers

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

Comments

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.