I found the following exercise problem in Gilbert Strang's Linear Algebra:
For four linear equations in two unknowns $x$ and $y$, the row picture shows four ____(A)______. The column picture is in ______(B)______ dimensional space. The equations have no solution unless the vector on the right hand is a combination of __(C)_________.
I tried to answer it in the following way: Let the four equations be $a_ix+b_iy+0v+0w=c_i$ where $i\in \{1,2,3,4\}$. Each of these equations represent a $3D$ plane in $4D$ space. This answers (A). (B) is of course "4 dimensional space" and (C) is "Columns on the left hand side".
Help needed in (A):
I doubt that what I have done above for (A) may be incorrect since the question specifically mentions that there are two unknowns $x$ and $y$ but I have assumed four $x,y,v,w$. Hence, (A) can also be 4 straight. lines?
But if I am wrong then why can't I assume the equations in forms above. If I write $x+y=1$, it represents a straight line in 2D and a 2D plane in 3D space. Hence it can be written as $x+y+0z=1$.
Please help me understand what should be in (A)? Thanks in advance.