SOLUTION: Please help solve the following. The instructions read "Solve each system by triangularizing the augmented matrix and using back-substitution. If the system is linearly dependent

Question 576079: Please help solve the following. The instructions read "Solve each system by triangularizing the augmented matrix and using back-substitution. If the system is linearly dependent, give the solution in terms of a parameter. If the system has coincident dependence, answer in set notation.
x + 3y +5z = 20
2x + 3y +4z = 16
x + 2y +3z = 12
I listed the matrix as
1 3 5 20
2 2 4 16
1 2 3 12
I used the row operation -2R1 + R2 > R2 to get the matrix to
1 3 5 20
0 -3 -6 -24
1 2 3 12
I then used row operation -1R1 + R3 > R3 to get the matrix to
1 3 5 20
0 -3 -6 -24
0 -1 -2 -8
I then used -1/3R2 + R3 > R3 to get the matrix to
1 3 5 20
0 -3 -6 -24
0 0 0 0
I know the system is linearly dependent but do understand how to solve from here. Please provide detail. Thank you. (also please let me know if I have made a mistake so far. I was given the answer of (p-4, -2P+8, p) but do not know how to come up with the answer with what I have so far.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
You're doing great. I'm going to finish up.

Row 2 has the values 0 -3 -6 -24, which means that -3y -6z = -24. Solve for y to get

-3y -6z = -24

-3y = 6z -24

y = (6z)/(-3) -24/(-3)

y = -2z + 8

Now if you let z = P (ie let z be the free variable), then y = -2P + 8. So that explains why the second coordinate is -2P+8 and the third is P

Now that you know that z = P and y = -2P+8, you can use this to find x in terms of the free variable P (or z). So use the first equation (you can use any equation that has x in it)

x + 3y +5z = 20

x + 3(-2P+8) +5P = 20

x -6P+24 +5P = 20

x -P+24 = 20

x -P = 20-24

x -P = -4

x = P-4

So the solutions are

x = P-4, y = -2P+8, and z = P

which form the ordered triple

(P-4, -2P+8, P)
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