- #1
0kelvin
- 50
- 5
\begin{cases}
x+ 2y - z + w - t = 0 \\
x - y + z + 3w - 2t = 0
\end{cases}
Add 1st to the 2nd:
$$2x + y + w - t = 0 \\
y = -2x -w + t = 0$$
Substitute y in the 1st:
##x + 2x + w - t + 3w - 2t + z = 0 \\
z = 3x - 4w + 3t##
Both z and y in terms of x,w,t. Writing using matrix form:
##\left[\begin{matrix}
x \\
y \\
z \\
w \\
t \\
\end{matrix}\right] =
x\left[\begin{matrix}
1 \\
-2 \\
-3 \\
0 \\
0 \\
\end{matrix}\right] +
w\left[\begin{matrix}
0 \\
-1 \\
-4 \\
1 \\
0 \\
\end{matrix}\right] +
t\left[\begin{matrix}
0 \\
1 \\
3 \\
0 \\
1 \\
\end{matrix}\right]
##
Solution ##x(1,-2,-3,0,0) + w(0,-1,-4,1,0) + t(0,1,3,0,1)##. I can isolate any two variables, but I have no idea if different solutions are in fact, part of the same set of solutions.
x+ 2y - z + w - t = 0 \\
x - y + z + 3w - 2t = 0
\end{cases}
Add 1st to the 2nd:
$$2x + y + w - t = 0 \\
y = -2x -w + t = 0$$
Substitute y in the 1st:
##x + 2x + w - t + 3w - 2t + z = 0 \\
z = 3x - 4w + 3t##
Both z and y in terms of x,w,t. Writing using matrix form:
##\left[\begin{matrix}
x \\
y \\
z \\
w \\
t \\
\end{matrix}\right] =
x\left[\begin{matrix}
1 \\
-2 \\
-3 \\
0 \\
0 \\
\end{matrix}\right] +
w\left[\begin{matrix}
0 \\
-1 \\
-4 \\
1 \\
0 \\
\end{matrix}\right] +
t\left[\begin{matrix}
0 \\
1 \\
3 \\
0 \\
1 \\
\end{matrix}\right]
##
Solution ##x(1,-2,-3,0,0) + w(0,-1,-4,1,0) + t(0,1,3,0,1)##. I can isolate any two variables, but I have no idea if different solutions are in fact, part of the same set of solutions.