Unsure How to Properly express my solution

frank789

Junior Member
Joined
Sep 16, 2017
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58
Ok working through a linear algebra problem set, I solve right to the end but my teacher has the solution in their key in a way I dont understand. Id rather not wait until monday as this is bugging me lol

Without further ado

x + 2y - 3z = 8
x + 3y - 5z = 11
2x + 5y - 8z = 19

so I made a matrix and the reduced row echelon form works out to (im 99.9 percent sure the matrix is right, if its not I apologize XD)
1 0 1 = 2
0 1 -2 = 3
0 0 0 = 0

so, I am asked to express solution as a vector. obviously one solution is (2,3,0) however the solution is written as
(x,y,z) = (2,3,0) + z(-1,2,1)

I understand the purpose of the z as this system should have infinite solutions due to the 0 row but I do not understand where the z(-1,2,1) comes from! Would be pumped if someone could point me in the right direction.
 
Your row-reduced matrix is correct. From there, it may be helpful to convert it back to a system of equations, to better see what's going on. Doing so gives three equations, one of which isn't terribly helpful, but I'll include it anyway for completeness' sake.

\(\displaystyle x + 0y + z = 2\)
\(\displaystyle 0x + y -2z = 3\)
\(\displaystyle 0x + 0y + 0z = 0\)

Since we've got a row with a single x and another row with a single y, getting information about z might be difficult. Additionally, with only two equations, we can never find a single value for all three variables... but what we can do is express two variables in terms of the third. Since we have good information about x and y, it makes sense to express those in terms of z. Let's do that now:

\(\displaystyle x = 2 - z\)
\(\displaystyle y = 3 + 2z\)

Excellent! This seems to be leading somewhere, because we have \(\displaystyle <x, y, z> = <2 - z, 3 + 2z, z> = <2, 3, 0> + <-z, 2z, z>\). Then we can factor out the z, resulting in exactly the given answer.
 
is there some way to pay you guys back for the help?/where do you all teach im transferring
 
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