I was solving the following problem:
Find the vector equation of the line representing the intersection of the planes
$2x + y - z = 10~$ and $~3x + 4y +2z = 30$
and although I eventually realized how to do it by setting up an augmented matrix, before I did, I tried two different methods and got two different answers and was just wondering what the equations I got using the incorrect method represent?
Method 1:
I made both equations $= 0$ by subtracting the constant and set them equal to each other. I simplified to get:
$x + 3y + 3z = 20$
and then using two parameters I converted it into a vector equation:
$ \left(\begin{array}{cc} x\\ y\\ z\\ \end{array}\right)= $ $ \left(\begin{array}{cc} 20\\ 0\\ 0\\ \end{array}\right) $ $+~\lambda \left(\begin{array}{cc} -3\\ 1\\ 0\\ \end{array}\right) $ $+~\mu \left(\begin{array}{cc} -3\\ 0\\ 1\\ \end{array}\right) $
So, I got an equation of a plane instead of a line. What is this plane? does it relate to anything or is it just a random plane because I did it incorrectly?
Method 2:
I solved for $y$ in each equation and then set them equal and converted to vector form to get:
$ \left(\begin{array}{cc} x\\ z\\ \end{array}\right) = $ $ \left(\begin{array}{cc} 2\\ 0\\ \end{array}\right) + \lambda~$ $ \left(\begin{array}{cc} 6/5\\ 1\\ \end{array}\right) $
Same questions as above? Although it isn't the provided answer (which I get by solving an augmented matrix of coefficients), does it have any relevance to anything or is it just a garbage answer?