Question

The following equations have infinitely many solutions. Give the right hand side of the vector form of the general solution, using a parameter such as s or t.

Answer 1

9x - 9y - 3z = 12
12x - 6y - z = 28
6x - 12y - 7z = 44

Answer 2

Is that in a Matrix?

Answer 3

Yeah, I'm assuming so... Of the form \[\left[\begin{matrix}9 & -9 & -3 &12\\ 12 & -6&-1&28\\6&-12&-7&44\end{matrix}\right]\]

Answer 4

Also, as an example, for the equations
x = y + 1, y = z + 1, z = x − 2
one correct answer is
[x, y, z] = [0, −1, −2] + t [1, 1, 1]

Answer 5

Could this be row reduction I think, you need zeros in bottom left hand corner Triangular matrix, so you can solve for z. Gaussian Elimination or Cramers Rule, does this sound right?

Answer 6

Yeah, I think so... I can get the matrix down to \[\left[\begin{matrix}1 & 0&1/6&10/3 \\ 0&1&1/2 & 2\\0&0&0&0\end{matrix}\right]\]but not sure what the answer would be... Thanks for the help :)

Answer 7

t3 = 1

t2 + 1/2t3 = 2
1 + 0.5 = 2
t2 = 0.5

t1 + 1/6t3 = 10/3
1+ 1/6 = 10/3
t1 =3(1/6)

z = 1
y = 0.5
x = 0.5

Matrix is linearly dependant infinite solutions exist. As soon as you get all zeros across the bottom row linearly dependant. I think this is right