### About Gaussian Elimination Two Variables:

We may prefer to solve a linear system using matrix methods. To perform this action, we can set up an augmented matrix, and then use row operations to place our matrix in row-echelon form or reduced-row echelon form. At this point, we will be able to find our solution for the system.

Test Objectives
• Demonstrate the ability to set up an augmented matrix
• Demonstrate the ability to place a matrix in row-echelon form
• Demonstrate the ability to place a matrix in reduced-row echelon form
• Demonstrate the ability to solve a linear system using matrix methods
Gaussian Elimination Two Variables Practice Test:

#1:

Instructions: solve each system.

$$a)\hspace{.2em}$$ $$2y=6 - 2x$$ $$0=33y - 21x + 63$$

$$b)\hspace{.2em}$$ $$4 - 7x=5y$$ $$0=-9y - 9x$$

#2:

Instructions: solve each system.

$$a)\hspace{.2em}$$ $$-3y + 9=5x$$ $$-12 - 9x=-4y$$

$$b)\hspace{.2em}$$ $$0=-12 + 6y - 7x$$ $$-1=x - y$$

#3:

Instructions: solve each system.

$$a)\hspace{.2em}$$ $$y + 5=-2x$$ $$-3x=5y - 24$$

#4:

Instructions: place in reduced-row echelon form

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}-3&-2&-6\\ -2&3&-4 \end{array}\right]$$

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}3&-2&18\\ -2&3&-12 \end{array}\right]$$

#5:

Instructions: place in reduced-row echelon form

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}5&3&15\\ 3&-4&9 \end{array}\right]$$

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}7&4&14\\ 1&-1&2 \end{array}\right]$$

$$c)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}5&-8&-3\\ 6&-9&0 \end{array}\right]$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.2em}(3,0)$$

$$b)\hspace{.2em}(2,-2)$$

#2:

Solutions:

$$a)\hspace{.2em}(0,3)$$

$$b)\hspace{.2em}(-6,-5)$$

#3:

Solutions:

$$a)\hspace{.2em}(-7,9)$$

#4:

Solutions:

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}1&0&2\\ 0&1&0 \end{array}\right]$$

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}1&0&6\\ 0&1&0 \end{array}\right]$$

#5:

Solutions:

$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}1&0&3\\ 0&1&0 \end{array}\right]$$

$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}1&0&2\\ 0&1&0 \end{array}\right]$$

$$c)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}1&0&9\\ 0&1&6 \end{array}\right]$$