We are given the following system of equations:

1. $7x - 4y + 7z = -5$
2. $6x - y + 10z = 3$
3. $5x - 4y - 4z = -7$

To solve this system, we can use either substitution, elimination, or matrix methods like Gaussian elimination. I'll use the elimination method to find the values of $x$, $y$, and $z$.

Step 1: Eliminate one variable (let's eliminate $y$).

From equation (1): 7x - 4y + 7z = -5 \tag{1}

From equation (2): 6x - y + 10z = 3 \tag{2}

From equation (3): 5x - 4y - 4z = -7 \tag{3}

Multiply equation (2) by 4 to align the coefficients of $y$ in equation (1) and (2):

$4(6x - y + 10z) = 4(3)$ This gives: 24x - 4y + 40z = 12 \tag{4}

Step 2: Subtract equation (1) from equation (4).

$(24x - 4y + 40z) - (7x - 4y + 7z) = 12 - (-5)$ This simplifies to: 17x + 33z = 17 \tag{5}

Step 3: Subtract equation (3) from equation (1).

$(7x - 4y + 7z) - (5x - 4y - 4z) = -5 - (-7)$ This simplifies to: 2x + 11z = 2 \tag{6}

Step 4: Solve the system of two equations with two unknowns ($x$ and $z$).

We now have:

1. $17x + 33z = 17$
2. $2x + 11z = 2$

Multiply the second equation by 17 to align the coefficients of $x$:

$17(2x + 11z) = 17(2)$ This gives: 34x + 187z = 34 \tag{7}

Step 5: Subtract equation (5) from equation (7).

$(34x + 187z) - (17x + 33z) = 34 - 17$ This simplifies to: $17x + 154z = 17$ Now subtract: $154z = 0$ Thus, $z = 0$.

Step 6: Substitute $z = 0$ into one of the previous equations.

Substitute $z = 0$ into equation (6): $2x + 11(0) = 2$ This simplifies to: $2x = 2 \implies x = 1$

Step 7: Solve for $y$.

Substitute $x = 1$ and $z = 0$ into one of the original equations. Let's use equation (1): $7(1) - 4y + 7(0) = -5$ This simplifies to: $7 - 4y = -5$ $-4y = -5 - 7$ $-4y = -12 \implies y = 3$

Final Solution:

The solution to the system of equations is: $x = 1, \quad y = 3, \quad z = 0$

Would you like further details on any step or have any questions?

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Here are 5 related questions to deepen your understanding:

1. How do you perform Gaussian elimination to solve systems of equations?
2. Can you apply substitution to this system of equations and check the result?
3. What happens if two equations are multiples of each other?
4. How do you solve systems of equations with three variables using matrices?
5. Can you use the method of determinants (Cramer's Rule) to solve this system?

Tip: When solving systems of equations, always check if you can simplify the system early on by eliminating variables, which can make the solution process faster.