To solve the system of equations using Gaussian Elimination, we start with the following system:

$\text{Equation 1: } 2x - y + 4z = -1$ $\text{Equation 2: } -4x + 2y - 8z = 2$

We will transform this system into an augmented matrix and then perform row operations to reach row-echelon form.

Step 1: Write the augmented matrix

The augmented matrix for the system is:

2 & -1 & 4 & | & -1 \\ -4 & 2 & -8 & | & 2 \end{pmatrix}$$ ### Step 2: Eliminate $$x$$ from the second row We want to make the coefficient of $$x$$ in the second row equal to 0. To do this, we'll multiply the first row by 2 and then add it to the second row: $$\text{Row 2} \rightarrow \text{Row 2} + 2 \times \text{Row 1}$$ Performing this: - Multiply Row 1 by 2: $$2 \times (2x - y + 4z) = (4x - 2y + 8z)$$ So, Row 1 becomes $$(4x - 2y + 8z = -2)$$. - Add this to Row 2: $$(-4x + 2y - 8z) + (4x - 2y + 8z) = (2 + (-2))$$ The result is: $$0x + 0y + 0z = 0$$ Thus, the second row becomes all zeros, which implies that the system has infinitely many solutions, and the second equation doesn’t provide new information. ### Step 3: The resulting matrix The augmented matrix now looks like: $$\begin{pmatrix} 2 & -1 & 4 & | & -1 \\ 0 & 0 & 0 & | & 0 \end{pmatrix}$$ ### Step 4: Back Substitution We are left with one equation: $$2x - y + 4z = -1$$ This equation has infinitely many solutions because we have two free variables ($$y$$ and $$z$$). ### General Solution: - Let $$y = t$$ and $$z = s$$ where $$t$$ and $$s$$ are free parameters. - Substitute into the first equation to express $$x$$: $$2x - t + 4s = -1$$ $$2x = t - 4s - 1$$ $$x = \frac{t - 4s - 1}{2}$$ Thus, the solution is: $$x = \frac{t - 4s - 1}{2}, \quad y = t, \quad z = s$$ Where $$t$$ and $$s$$ are any real numbers. --- Would you like to go over any specific details or steps? Here are 5 related questions to expand on this: 1. How would we solve this system using substitution instead of Gaussian elimination? 2. What is the condition for a system of linear equations to have infinite solutions? 3. How would you apply Gaussian elimination to a system with three variables? 4. How does Gaussian elimination compare to other methods like Cramer's rule or matrix inversion? 5. What are some common pitfalls to avoid when using Gaussian elimination? **Tip**: Always check for inconsistencies in the augmented matrix. If you encounter a row like $$[0, 0, 0 | b]$$ where $$b \neq 0$$, the system has no solution.