You provided a system of three equations:

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

Solving the System:

We can solve this system of linear equations using methods like substitution, elimination, or matrix methods (like Gaussian elimination). I will proceed with the elimination method:

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Step 1: Add Equation 2 and Equation 3

From equations 2 and 3:

$(3x - 5y + 3z) + (-5x + 6y - 3z) = (-1 + 5)$ This simplifies to:

-2x + y = 4 \tag{4}

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Step 2: Multiply Equation 2 by 2 and add to Equation 1

We want to eliminate $z$. Multiply equation 2 by 2 and add it to equation 1:

$2(3x - 5y + 3z) = 2(-1)$ $6x - 10y + 6z = -2$

Now add it to equation 1:

$(6x + 4y - 3z) + (6x - 10y + 6z) = (-10 - 2)$

This simplifies to:

12x - 6y + 3z = -12 \tag{5}

Now we have two simpler equations to solve further.

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Do you want me to continue solving this system and find the values of $x$, $y$, and $z$? Let me know if you want more details or explanation.

Related questions:

1. What are some common methods for solving systems of equations?
2. How does Gaussian elimination work for larger systems?
3. Can we use substitution here to solve the system more easily?
4. How would a system like this look in matrix form?
5. What are the advantages of using elimination vs substitution?

Tip: Always double-check the consistency of signs in the equations during elimination to avoid errors.