Science:Math Exam Resources/Courses/MATH152/April 2017/Question B 01 (c)/Solution 1

From (b) the augmented matrix is:

${\begin{bmatrix}1&0&-3&|0\\1&-2&-2&|0\\1&1&1&|0.25\\\end{bmatrix}}$

Then we follow the steps for Gaussian elimination: Multiply the 1st row by $-1$ and add it to the 2nd and 3rd row:

${\begin{bmatrix}1&0&-3&|0\\0&-2&1&|0\\0&1&4&|0.25\\\end{bmatrix}}$

Then switch the 2nd and 3rd rows with each other

${\begin{bmatrix}1&0&-3&|0\\0&1&4&|0.25\\0&-2&1&|0\\\end{bmatrix}}$

Then multiply the 2nd row by $2$ and add it to 3rd row:

${\begin{bmatrix}1&0&-3&|0\\0&1&4&|0.25\\0&0&9&|0.5\\\end{bmatrix}}$

Divide the 3rd by 9:

${\begin{bmatrix}1&0&-3&|0\\0&1&4&|0.25\\0&0&1&|1/18\\\end{bmatrix}}$

Last step is to cancel values above 1 in the 3 third column, therefore the final form of the augmented matrix in upper echeleon form is:

${\begin{bmatrix}1&0&0&|1/6\\0&1&0&|1/36\\0&0&1&|1/18\\\end{bmatrix}}$

Answer: $\color {blue}x_{1}={\frac {1}{6}}Room/Hour,\,x_{3}={\frac {1}{18}}Room/Hour,\,x_{2}={\frac {1}{36}}Room/Hour$