gaussian elimination:3x-4y-6z=-19 -3x+2y-4z=-21 5x-y+2z=0 find x,y,z

Question

given the system,change to matrix form.then use Gaussian elimination to get it to reduced Row Echelon Form.Then state the solution:

3x-4y-6z=-19

-3x+2y-4z=-21

5x-y+2z=0

Answer 1

This would be staggeringly tedious to show on this platform.  But, I can give you the starting matrix, and the row operations to resolve it.  I have to confess there is a shorter series of row operations that get you to the answer, but I elected this method to avoid fractional values.  (Just because of the tedium and cramped writing involved.  This has 10 row operations, the optimal method uses 9)

 

⌈3  -4  -6  | -19⌉

|-3  2  -4  | -21|

⌊ 5  -1   2 |    0⌋

 

R1 + R2 → R2

R3 - R1 → R3

R1 - R3 → R1

R3 - 2R1 → R3

(-1/2)R2 → R2

R3 - 17R2 → R3

(-1/49)R3 → R3

R2 - 5R3 → R2

R1 + 14R3 → R1

R1 + 7R2 → R1

 

which will give a final matrix of

 

⌈1  0  0  | -3⌉

|0  1  0  | -5|

⌊0  0  1  |  5⌋

 

so x = -3, y = -5, and z = 5

Answer 2

(-3, -5, 5)...USING RREF command on TI-83 calculator (MATRIX menu).

 

If you must do it by hand, then I assume that there are examples and pointers in your book or class material.  You will do ROW OPERATIONS.  I'm sure not going to type all of the steps (and numbers in rows) for you...You could start by adding row 2's numbers to row 3, and then ...  basically, your goal is to create last row as 0 0 1 5, and second row 0 1 0 -5, etc.

 

You do have to be careful, especially when there comes a time to divide a row, thus introducing fractions.  (If you can avoid fractions, you avoid a lot of potential errors.)