You're correct. The matrix transformation I performed didn't lead to a correct upper triangular matrix. Let's go through the correct steps to achieve the upper triangular form using the Gauss Elimination method.

Step 1: Write the system of equations in standard form

The equations are already in standard form:

1. $3x + 4y + 5z = 18$
2. $2x - y + 8z = 13$
3. $5x - 2y + 7z = 20$

Step 2: Write the augmented matrix

The augmented matrix corresponding to the system is:

3 & 4 & 5 & | & 18 \\ 2 & -1 & 8 & | & 13 \\ 5 & -2 & 7 & | & 20 \\ \end{pmatrix}$$ ### Step 3: Perform row operations to get an upper triangular matrix #### a) Make the first element of the first column a pivot (if necessary): The first row is fine as is, so let's eliminate the elements below it in the first column. - **Eliminate the element in the second row, first column:** $$R2 \rightarrow R2 - \frac{2}{3}R1$$ Calculation: $$\begin{pmatrix} 3 & 4 & 5 & | & 18 \\ 0 & -\frac{11}{3} & \frac{14}{3} & | & 1 \\ 5 & -2 & 7 & | & 20 \\ \end{pmatrix}$$ - **Eliminate the element in the third row, first column:** $$R3 \rightarrow R3 - \frac{5}{3}R1$$ Calculation: $$\begin{pmatrix} 3 & 4 & 5 & | & 18 \\ 0 & -\frac{11}{3} & \frac{14}{3} & | & 1 \\ 0 & -\frac{26}{3} & \frac{4}{3} & | & -10 \\ \end{pmatrix}$$ #### b) Make the pivot in the second column: - **Eliminate the element in the third row, second column:** We now eliminate the element below the pivot $$y$$ in the third row by the following operation: $$R3 \rightarrow R3 - \frac{-26/3}{-11/3}R2$$ Simplification gives: $$R3 \rightarrow R3 + 2 \times R2$$ Calculation: $$\begin{pmatrix} 3 & 4 & 5 & | & 18 \\ 0 & -\frac{11}{3} & \frac{14}{3} & | & 1 \\ 0 & 0 & -6 & | & -8 \\ \end{pmatrix}$$ Now we have the upper triangular form. ### Step 4: Back substitution 1. **Solve for $$z$$:** $$-6z = -8 \quad \Rightarrow \quad z = \frac{4}{3}$$ 2. **Solve for $$y$$:** Using the second row: $$-\frac{11}{3}y + \frac{14}{3} \times \frac{4}{3} = 1$$ Simplifying: $$-\frac{11}{3}y + \frac{56}{9} = 1 \quad \Rightarrow \quad y = \frac{53}{33}$$ 3. **Solve for $$x$$:** Using the first row: $$3x + 4 \times \frac{53}{33} + 5 \times \frac{4}{3} = 18$$ Simplifying: $$x = \frac{14}{9}$$ ### Conclusion The correct solutions are: - $$x = \frac{14}{9}$$ - $$y = \frac{53}{33}$$ - $$z = \frac{4}{3}$$ Would you like more details on these steps? Here are some related questions: 1. What does it mean for a matrix to be in upper triangular form? 2. How can you check if a system of equations has a unique solution? 3. How does pivoting help in the Gauss Elimination method? 4. Can the Gauss Elimination method be used on non-square matrices? 5. What are the conditions for a system of linear equations to have no solutions? **Tip:** Always keep track of each row operation to avoid mistakes when forming the upper triangular matrix.