Determine whether there is a unique solution, no solution, or an infinite set of solutions with one or two parameters.

$\displaystyle x+y-2z=1$

$\displaystyle 3x+y-4z=2$

$\displaystyle 4x+2y-6z=5$

I'm supposed to use an augmented matrix to solve this.

$\displaystyle \left(\begin{array}{ccc}1 & 1 & -2\\3 & 1 & -4\\4 & 2 & -6\end{array}\right)$ (I don't know how to make the augmented part of the matrix.)

$\displaystyle r_1+r_2\rightarrow\left(\begin{array}{ccc}4&2&-6\\3&1&-4\\4&2&-6\end{array}\right)$

$\displaystyle r_1-r_3\rightarrow\left(\begin{array}{ccc}0&0&0\\3&1&-4\\4&2&-6\end{array}\right)$

When reducing the matrix, I get a row of zeros, so this means that there is a line of intersection for the planes, so an infinite set of solutions dependent on one parameter.

But the answer says that there is no solution.

I found the line to be: $\displaystyle x=\lambda,y=-\lambda,z=\lambda-\frac{1}{2}$

Thanks!

PS. I hope this is the right section. Sorry Mods if its not