Question

Use gauss-jordan row reduction to solve the given system of equations. hint [see examples 1-6.] (if there is no solution, enter no solution. if the system is dependent, express your answer in terms of x, where y = y(x), z = z(x), and w = w(x).) x + y + 5z = −2 y + 6z + w = 3 x + 3y + 11z + 2w = 1 x + y + 5z + w = −2

Answer 1

The augmented matrix for the system can be written as ...
$\left[\begin{array}{ccccc}1&1&5&0&-2\\0&1&6&1&3\\1&3&11&2&1\\1&1&5&1&-2\end{array}\right]$
where the order of the variables is x, y, z, w. After subtracting the first row from the 3rd and 4th rows, this becomes ...
$\left[\begin{array}{ccccc}1&1&5&0&-2\\0&1&6&1&3\\0&2&6&2&3\\0&0&0&1&0\end{array}\right]$
Then subtracting twice the second row from the third, you have ...
$\left[\begin{array}{ccccc}1&1&5&0&-2\\0&1&6&1&3\\0&0&-6&0&-3\\0&0&0&1&0\end{array}\right]$

The 4th row tells you w = 0.
The third row tells you z = -3/-6 = 1/2.
Substituting these values into the equation for y gives y +6/2 +0 = 3, so y = 0.
Finally, substituting into the equation for x, we get x +0 +5/2 = -2, so x = -9/2.

The solution is (x, y, z, w) = (-9/2, 0, 1/2, 0).