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Problem 11

# Use Gauss-Jordan row reduction to solve the given systems of equation. We suggest doing some by hand, and others using technology. HINT [See Examples 1-6.] $$\begin{array}{r} x+y=0 \\ 3 x-y=1 \\ x-y=-1 \end{array}$$

Expert verified
The unique solution to the given system of equations using Gauss-Jordan row reduction is: $\begin{array}{r} x = 1 \\ y = -1 \\ z = 1 \end{array}$
See the step by step solution

## Step 1: Write the augmented matrix

Write the given system of equations in the form of an augmented matrix: $\left[ \begin{array}{ccc|c} 1 & 1 & 0 \\ 3 & -1 & 1 \\ 1 & -1 & -1 \end{array} \right]$

## Step 2: Perform row operations to reach the RREF

We will perform the following row operations to reach the RREF: 1. Add -3 times Row 1 to Row 2. 2. Add -1 times Row 1 to Row 3. 3. Multiply Row 2 by -1/4. 4. Add Row 2 to Row 3. 5. Multiply Row 3 by 1/3. Let's now perform these operations: 1. Add -3 times Row 1 to Row 2: $\left[ \begin{array}{ccc|c} 1 & 1 & 0 & 0 \\ 0 & -4 & 1 & 1 \\ 1 & -1 & -1 & -1 \end{array} \right]$ 2. Add -1 times Row 1 to Row 3: $\left[ \begin{array}{ccc|c} 1 & 1 & 0 & 0 \\ 0 & -4 & 1 & 1 \\ 0 & -2 & -1 & -1 \end{array} \right]$ 3. Multiply Row 2 by -1/4: $\left[ \begin{array}{ccc|c} 1 & 1 & 0 & 0 \\ 0 & 1 & -1/4 & -1/4 \\ 0 & -2 & -1 & -1 \end{array} \right]$ 4. Add Row 2 to Row 3: $\left[ \begin{array}{ccc|c} 1 & 1 & 0 & 0 \\ 0 & 1 & -1/4 & -1/4 \\ 0 & 0 & -3/4 & -3/4 \end{array} \right]$ 5. Multiply Row 3 by 1/3: $\left[ \begin{array}{ccc|c} 1 & 1 & 0 & 0 \\ 0 & 1 & -1/4 & -1/4 \\ 0 & 0 & 1 & 1 \end{array} \right]$

## Step 3: Interpret the RREF matrix

Now that we have reached the RREF, let's interpret the result: $\begin{array}{r} x + y = 0\\ y - (1/4)z = -1/4 \\ z = 1 \end{array}$ Now we can solve for the other variables using back-substitution: 1. Substitute z into the second equation: $$y-(1/4)(1)=-1/4$$ so $$y=-1$$ 2. Substitute y into the first equation: $$x+(-1)=0$$ so $$x=1$$ Finally, we have found the unique solution to the given system of equations: $\begin{array}{r} x = 1 \\ y = -1 \\ z = 1 \end{array}$

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