I am trying to find the general solution to this system of equations using an augmented matrix, and then using the gauss reduction technique, but i cant seem to get it into row echelon form no matter what i try. The system is:
4w - 5x - y + 21z = 17
-w + 3x + 4y - 2z = -2
w - 2x - 2y + 4z = 3
3w - 8x - 8y + 6z = 11I have tried everything i can think of, but i cant seem to figure it out. If someone could please give me some pointers i would be immensely grateful.
Thanks in advance Timothy
$\endgroup$ 01 Answer
$\begingroup$The following should be manageable enough for you to follow (note that you need to obtain the reduced echelon form of the system of equations in order to determine the general solution): \begin{align} \begin{bmatrix}4&-5&-1&21&17\\-1&3&4&-2&-2\\1&-2&-2&4&3\\3&-8&-8&6&11\end{bmatrix}&\sim \begin{bmatrix}1&-2&-2&4&3\\-1&3&4&-2&-2\\4&-5&-1&21&17\\3&-8&-8&6&11\end{bmatrix}\\[1em] &\sim\begin{bmatrix}1&-2&-2&4&3\\0&1&2&2&1\\0&3&7&5&5\\0&-2&-2&-6&2\end{bmatrix}\\[1em] &\sim\begin{bmatrix}1&-2&-2&4&3\\0&1&2&2&1\\0&0&1&-1&2\\0&0&2&-2&4\end{bmatrix}\\[1em] &\sim\begin{bmatrix}1&-2&-2&4&3\\0&1&2&2&1\\0&0&1&-1&2\\0&0&0&0&0\end{bmatrix}\\[1em] &\sim\begin{bmatrix}1&-2&0&2&7\\0&1&0&4&-3\\0&0&1&-1&2\\0&0&0&0&0\end{bmatrix}\\[1em] &\sim\begin{bmatrix}1&0&0&10&1\\0&1&0&4&-3\\0&0&1&-1&2\\0&0&0&0&0\end{bmatrix}\\[1em] \end{align} Thus, your general solution is given by \begin{cases} w &= 1-10z\\ x &= -3-4z\\ y &= 2+z\\ z & \text{is free} \end{cases}
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