Need help with matrix algebra, should be simple in theory?

Question

Find the line of intersection of the following planes:

3x +2y +z = -1
2x -y + 4z = 5

To my understanding, in theory this should be a simple augmented matrix that just needs to be row reduced. But no matter how I try to do it, I get the following answer:

x = -9t/7 + 9/7
y = -10t/7 - 17/7
z = t

And the correct answer is supposedly:

x = 9t
y = -10t -1
z = -7t +1

I'm extremely frustrated here, can anyone lend a hand?

Hosam · Accepted Answer

The reduced-row echelon form of the augmented matrix is

1 ... 0...9/7......9/7
0.....1...-10/7....-17/7

So if you z = t , then
y = -17/7 + 10/7 t
x = 9/7 - 9/7 t

which is your answer (except for the sign of 10t/7 in the equation for y)

But to avoid fractions, we can choose z = -7t+1, then
y = -17/7 + 10/7(-7t+1) = -17/7 - 10 t + 10/7 = -10 t - 1, and
x = 9/7 - 9/7 (-7t+1) = 9 t

Which is the book's answer.  In summary, both answer are correct (they are equivalent).

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Need help with matrix algebra, should be simple in theory?

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