To construct the augmented matrix for the given system of equations, we first need to rewrite the system in a standard linear form. The standard form is $ax + by + cz = d$, where $a$, $b$, $c$, and $d$ are constants. Let's rearrange each equation accordingly:

System of Equations

1. $5 + 7x = y + 4z$
2. $-2(x - 2y) = 0$
3. $7x - y = 2(x - 5z)$

Step-by-Step Solution

1. Rearrange the first equation:

$5 + 7x = y + 4z$ Subtract $y + 4z$ from both sides: $7x - y - 4z = -5$

2. Simplify the second equation:

$-2(x - 2y) = 0$ Distribute the $-2$: $-2x + 4y = 0$

3. Simplify the third equation:

$7x - y = 2(x - 5z)$ Distribute the $2$: $7x - y = 2x - 10z$ Subtract $2x$ from both sides: $5x - y + 10z = 0$

Final System

7x - y - 4z = -5 \\ -2x + 4y = 0 \\ 5x - y + 10z = 0 \end{cases}$$ ### Augmented Matrix The augmented matrix for this system is constructed by aligning the coefficients of $$x$$, $$y$$, $$z$$, and the constants on the right side: $$\begin{bmatrix} 7 & -1 & -4 & | & -5 \\ -2 & 4 & 0 & | & 0 \\ 5 & -1 & 10 & | & 0 \end{bmatrix}$$ This matrix represents the coefficients of the variables and the constants from the system of equations. Would you like more details or have any questions about this process? Here are 5 related questions: 1. How do you determine if an augmented matrix has a unique solution? 2. What are the steps to perform row reduction on an augmented matrix? 3. How do you find the inverse of a matrix if it exists? 4. What does it mean if an augmented matrix has a row of all zeros? 5. How can an augmented matrix be used to solve a system of equations with more than three variables? **Tip:** When constructing an augmented matrix, ensure each equation is arranged with the variables aligned and in the same order. This consistency is crucial for accurate matrix formation.