About Gaussian Elimination Three 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.
$$a)\hspace{.2em}$$ $$x + 9y - 5z=-32$$ $$-3x - 5y - 5z=-10$$ $$-2x - 7y + z=13$$
$$b)\hspace{.2em}$$ $$7x - 8y + z=-37$$ $$2x - y + 5z=-2$$ $$-2x + y - 5z=1$$
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#2:
Instructions: solve each system.
$$a)\hspace{.2em}$$ $$-10x - 9y + 2z=20$$ $$-5x + 6y - 5z=-11$$ $$-3x - 3y - z=19$$
$$b)\hspace{.2em}$$ $$-7x - 4y + 10z=1$$ $$5x + 10y - 2z=-19$$ $$5x + 2y - 7z=3$$
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#3:
Instructions: solve each system.
$$a)\hspace{.2em}$$ $$10x - y + 4z=-2$$ $$2x - y - 5z=-40$$ $$3x + 3y - 7z=-30$$
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#4:
Instructions: place in reduced-row echelon form.
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{ccc|c}-7&4&5&24\\ 8&1&1&-31\\6&-9&-7&-49 \end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{ccc|c}-4&2&5&11\\ 7&-1&5&-13\\-6&-8&4&46 \end{array}\right]$$
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#5:
Instructions: place in reduced-row echelon form.
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{ccc|c}-1&7&7&38\\ 6&-9&6&-48\\3&-4&1&-24 \end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{ccc|c}-2&6&3&-38\\ 1&-5&3&-19\\1&-3&1&-1 \end{array}\right]$$
$$c)\hspace{.2em}$$ $$\left[ \begin{array}{ccc|c}-6&-3&3&42\\ -7&-2&4&46\\-2&-7&5&44 \end{array}\right]$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.2em}(5,-3,2)$$
$$b)\hspace{.2em}No \hspace{.2em}Solution$$
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#2:
Solutions:
$$a)\hspace{.2em}(2,-6,-7)$$
$$b)\hspace{.2em}(5,-4,2)$$
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#3:
Solutions:
$$a)\hspace{.2em}(-2,6,6)$$
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#4:
Solutions:
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{ccc|c}1&0&0&-4\\ 0&1&0&9\\ 0&0&1&-8 \end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{ccc|c}1&0&0&-3\\ 0&1&0&-3\\ 0&0&1&1 \end{array}\right]$$
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#5:
Solutions:
$$a)\hspace{.2em}$$ $$\left[ \begin{array}{ccc|c}1&0&0&-3\\ 0&1&0&4\\ 0&0&1&1 \end{array}\right]$$
$$b)\hspace{.2em}$$ $$\left[ \begin{array}{ccc|c}1&0&0&10\\ 0&1&0&1\\ 0&0&1&-8 \end{array}\right]$$
$$c)\hspace{.2em}$$ $$\left[ \begin{array}{ccc|c}1&0&0&-4\\ 0&1&0&-3\\ 0&0&1&3 \end{array}\right]$$
