We are solving a system of three equations with three variables ($x$, $y$, and $z$):

1. $-3x + 9y - 5z = -5$
2. $-3x + 2y - 3z = -9$
3. $3x + 6y + 4z = 7$

Steps:

1. Simplify and organize:

The equations are already organized, so let’s start solving using the substitution or elimination method.

2. Eliminate $x$ between the first two equations:

Subtract equation 2 from equation 1:

$(-3x + 9y - 5z) - (-3x + 2y - 3z) = -5 - (-9)$

Simplify:

$7y - 2z = 4$

3. Eliminate $x$ between the first and third equations:

Add equation 1 and equation 3:

$(-3x + 9y - 5z) + (3x + 6y + 4z) = -5 + 7$

Simplify:

$15y - z = 2$

4. Solve the system of two equations in $y$ and $z$:

Now solve the system:

1. $7y - 2z = 4$
2. $15y - z = 2$

From the second equation, solve for $z$ in terms of $y$:

$z = 15y - 2$

Substitute $z = 15y - 2$ into the first equation:

$7y - 2(15y - 2) = 4$

Simplify:

$7y - 30y + 4 = 4$

$-23y = 0$

$y = 0$

5. Solve for $z$:

Substitute $y = 0$ into $z = 15y - 2$:

$z = 15(0) - 2 = -2$

6. Solve for $x$:

Substitute $y = 0$ and $z = -2$ into any original equation, such as equation 2:

$-3x + 2(0) - 3(-2) = -9$

$-3x + 6 = -9$

$-3x = -15$

$x = 5$

Final Solution:

$x = 5, \, y = 0, \, z = -2$

Would you like detailed verification or have any specific questions?

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Relative Questions:

1. How do substitution and elimination methods differ in solving systems of equations?
2. What is the geometric interpretation of solving three equations with three variables?
3. How does one verify the solution of a system of equations?
4. Can you solve systems of equations with dependent or inconsistent solutions?
5. What are some real-world applications of solving such systems of equations?

Tip:

Always double-check your solution by substituting the values back into the original equations.