To determine if the system is consistent, we can write it in augmented matrix form and then perform Gaussian elimination or row reduction to see if we encounter any contradictions.

The system of equations is:

1. $x_1 - 2x_4 = -3$
2. $2x_2 + 2x_3 = 0$
3. $x_3 + 3x_4 = 1$
4. $-2x_1 + 3x_2 + 2x_3 + x_4 = 5$

The corresponding augmented matrix is:

1 & 0 & 0 & -2 & | & -3 \\ 0 & 2 & 2 & 0 & | & 0 \\ 0 & 0 & 1 & 3 & | & 1 \\ -2 & 3 & 2 & 1 & | & 5 \end{pmatrix}$$ Now, we'll use row operations to reduce this matrix. Let's proceed step by step: ### Step 1: Eliminate $$x_1$$ from the fourth row We add 2 times row 1 to row 4: $$R_4 \to R_4 + 2R_1$$ This gives us: $$\begin{pmatrix} 1 & 0 & 0 & -2 & | & -3 \\ 0 & 2 & 2 & 0 & | & 0 \\ 0 & 0 & 1 & 3 & | & 1 \\ 0 & 3 & 4 & -3 & | & -1 \end{pmatrix}$$ ### Step 2: Eliminate $$x_2$$ from the fourth row Next, we subtract $$\frac{3}{2}$$ times row 2 from row 4 to eliminate the $$x_2$$-term: $$R_4 \to R_4 - \frac{3}{2} R_2$$ This gives us: $$\begin{pmatrix} 1 & 0 & 0 & -2 & | & -3 \\ 0 & 2 & 2 & 0 & | & 0 \\ 0 & 0 & 1 & 3 & | & 1 \\ 0 & 0 & 1 & -3 & | & -1 \end{pmatrix}$$ ### Step 3: Eliminate $$x_3$$ from the fourth row Now, subtract row 3 from row 4 to eliminate the $$x_3$$-term: $$R_4 \to R_4 - R_3$$ This gives us: $$\begin{pmatrix} 1 & 0 & 0 & -2 & | & -3 \\ 0 & 2 & 2 & 0 & | & 0 \\ 0 & 0 & 1 & 3 & | & 1 \\ 0 & 0 & 0 & -6 & | & -2 \end{pmatrix}$$ ### Step 4: Simplify the fourth row Divide the fourth row by $$-6$$ to simplify it: $$R_4 \to \frac{1}{-6} R_4$$ This gives: $$\begin{pmatrix} 1 & 0 & 0 & -2 & | & -3 \\ 0 & 2 & 2 & 0 & | & 0 \\ 0 & 0 & 1 & 3 & | & 1 \\ 0 & 0 & 0 & 1 & | & \frac{1}{3} \end{pmatrix}$$ ### Step 5: Check for consistency Now, we observe the system. There is no row with all zeros except for the last column, which would indicate a contradiction. The system does not have any row like $$0 = \text{non-zero number}$$, meaning there is no inconsistency. Thus, the system **is consistent**. --- Would you like to go over the row operations in more detail or have any other questions? Here are some related questions: 1. How would you solve this system if it were a homogenous system (where the right-hand side was all zeros)? 2. What is the rank of the augmented matrix for this system? 3. How would the system change if the second equation were $$2x_2 + 2x_3 = 5$$? 4. Can you identify the free variables in the system from the reduced matrix? 5. What would happen if there were a row of the form $$0 = -5$$ in the row-reduced matrix? **Tip:** Always perform row reduction carefully, and check for contradictions at each step to determine consistency.