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
#1:
Instructions: solve each system.
$$a)\hspace{.2em}$$ $$2y=6 - 2x$$ $$0=33y - 21x + 63$$
$$b)\hspace{.2em}$$ $$4 - 7x=5y$$ $$0=-9y - 9x$$
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#2:
Instructions: solve each system.
$$a)\hspace{.2em}$$ $$-3y + 9=5x$$ $$-12 - 9x=-4y$$
$$b)\hspace{.2em}$$ $$0=-12 + 6y - 7x$$ $$-1=x - y$$
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#3:
Instructions: solve each system.
$$a)\hspace{.2em}$$ $$y + 5=-2x$$ $$-3x=5y - 24$$
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#4:
Instructions: place in reduced-row echelon form
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}-3&-2&-6\\ -2&3&-4 \end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}3&-2&18\\ -2&3&-12 \end{array}\right]$$
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#5:
Instructions: place in reduced-row echelon form
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}5&3&15\\ 3&-4&9 \end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}7&4&14\\ 1&-1&2 \end{array}\right]$$
$$c)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}5&-8&-3\\ 6&-9&0 \end{array}\right]$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}(3,0)$$
$$b)\hspace{.2em}(2,-2)$$
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#2:
Solutions:
$$a)\hspace{.2em}(0,3)$$
$$b)\hspace{.2em}(-6,-5)$$
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#3:
Solutions:
$$a)\hspace{.2em}(-7,9)$$
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#4:
Solutions:
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}1&0&2\\ 0&1&0 \end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}1&0&6\\ 0&1&0 \end{array}\right]$$
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#5:
Solutions:
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}1&0&3\\ 0&1&0 \end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}1&0&2\\ 0&1&0 \end{array}\right]$$
$$c)\hspace{.2em}$$ $$\left[ \begin{array}{cc|c}1&0&9\\ 0&1&6 \end{array}\right]$$
