Definition: A $3 \times 3$ system of equations is 3 equations in 3 unknowns.
Example: $\left\{ \begin{array}{c}x+y-2z=3\\2x+3y+2z=-1\\3x-y-3z=2\\\end{array} \right\}.$
Solving $3 \times 3$ Systems Algebraically
Good News: the techniques we used to solve $2 \times 2$ systems also work for $3 \times 3$ systems.
Example: Solve the system of equations $$\left\{ \begin{array}{c}x+y-2z=3\\2x+3y+2z=-1\\3x-y-3z=2\\\end{array} \right\}.$$
Algebraic Process
- Step 1. Write each equation in form $Ax + By + Cz = D.$
- Step 2. Select a pair of equations and use the substitution method or the addition method to eliminate one of the variables.
- Step 3. Repeat Step 2 with a different pair of equations. Be sure to eliminate the same variable as in Step 2.
- Step 4. Solve the resulting $2 \times 2$ system of equations.
- Step 5. Substitute the values from Step 4 into one of the original equations to solve for the third variable.
- Step 6. Does this solution check in all three of the original equations?
Matrices as Representations of Systems: We can also represent $3 \times 3$ systems as augmented matrices.
Example: We can represent the system $$\left\{ \begin{array}{c}x+y-2z=3\\2x+3y+2z=-1\\3x-y-3z=2\\\end{array} \right\}$$ in matrix form $$\left[ \begin{array}{ccc|c} 1 & 1 & -2 & 3 \\ 2 & 3 & 2 & -1 \\ 3 & -1 & -3 & 2 \\ \end{array} \right].$$
Elementary Row Operations
Algebraic Operation
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Elementary Row Operation
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Solving Systems
Example: Now solve the system of equations performing "elementary row operations" on its augmented matrix. $$\left\{ \begin{array}{c}x+y-2z=3\\2x+3y+2z=-1\\3x-y-3z=2\\\end{array} \right\}$$
So, As You Can See...
If you can massage the augmented matrix of a system into the form $$\left[\begin{array}{ccc|c} 1 & 0 & 0 & \mbox{number}_1 \\ 0 & 1 & 0 & \mbox{number}_2 \\ 0 & 0 & 1 & \mbox{number}_3 \\ \end{array}\right]$$ by elementary row operations, you have effectively solved the system. I.e., $$\begin{array}{c} x=\mbox{number}_1 \\ y=\mbox{number}_2 \\ z=\mbox{number}_3\\ \end{array}$$
A Method for Massaging Matrices into the Form You Want
Two Unique Examples: Solve the following systems by performing row operations on their respective matrix forms. $$\left\{ \begin{array}{ccc} x+4y+5z=-2 \\ y+2z=-1 \\ -5y-10z=5 \\ \end{array} \right\}.$$ $$\left\{ \begin{array}{ccc} x+8y+2z=20 \\ 11y+z=28 \\ -22y-2z=-55 \\ \end{array} \right\}.$$
Vocab: The "end-forms" we have seen in the previous 3 examples after performing row operations in the manner described in our "Massage Method" are all called reduced row echelon form.
$\left[\begin{array}{ccc|c} 1 & 0 & 0 & -\frac{10}{27} \\ 0 & 1 & 0 & \frac{7}{9} \\ 0 & 0 & 1 & -\frac{35}{27} \\ \end{array}\right],$ $\left[\begin{array}{ccc|c} 1 & 0 & -3 & 2 \\ 0 & 1 & 2 & -1 \\ 0 & 0 & 0 & 0 \\ \end{array}\right],$ $\left[\begin{array}{ccc|c} 1 & 0 & 0 & -2 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 0 & 1 \\ \end{array}\right]$
Using Technology
As you can imagine, for larger systems the above process would be an onerous task. Even $3 \times 3$ systems aren't exactly a walk in the park. However, for a computer, it's child's play. That's why we use technology.
You may use either your TI-84 Calculator, or other technology for solving these systems. For example, a basic Google search gives us the following online app: mathportal.org
Important Note: You will need to figure out a calculator-based option if you want to use technology to solve systems on the final exam.
Application: A zookeeper named mixes three foods, the contents of which are described in the following table. How many grams of each food are needed to produce a mixture with 133 g of fat, 494 g of protein, and 1698 g of carbohydrates? Round your answer to the nearest hundredth. $$\begin{array}{l|l|l|l} \hline \mbox{} & A & B & C \\ \hline \mbox{Fat (%)} & 9 & 5 & 4 \\ \hline \mbox{Protein (%)} & 15 & 18 & 16 \\ \hline \mbox{Carbohydrates (%)} & 65 & 60 & 55 \\ \hline \end{array}$$