Explanation:

To solve the system of equations:

$$

\begin{cases}

3y + 4z - 6x = 0 \\

2x - 4y + z = 0 \\

5z - 8y - x = 0

\end{cases}

$$

We can use the **elimination method** to eliminate one variable at a time and then solve for the remaining variables.

First, we can eliminate **x** by adding the first and second equations, and subtracting the third equation from the second equation. This gives us:

$$

\begin{cases}

-7y + 5z = 0 \\

-3y - 4z = 0

\end{cases}

$$

Next, we can eliminate **y** by multiplying the first equation by 3 and adding it to the second equation. This gives us:

$$

-11z = 0

$$

Solving for **z**, we get:

$$

z = 0

$$

Substituting **z = 0** into the first equation, we get:

$$

-7y = 0

$$

Solving for **y**, we get:

$$

y = 0

$$

Substituting **y = 0** and **z = 0** into the original equations, we get:

$$

\begin{cases}

-6x = 0 \\

2x = 0 \\

-x = 0

\end{cases}

$$

Solving for **x**, we get:

$$

x = 0

$$

Therefore, the solution to the system of equations is:

$$

(x, y, z) = (0, 0, 0)

$$

This is different from the answer you provided, which is:

$$

(x, y, z) = (-5, -9, 10)

$$

To check if this answer is correct, we can substitute it into the original equations and see if they are satisfied. This gives us:

$$

\begin{cases}

3(-9) + 4(10) - 6(-5) = 0 \\

2(-5) - 4(-9) + 10 = 0 \\

5(10) - 8(-9) - (-5) = 0

\end{cases}

$$

Simplifying, we get:

$$

\begin{cases}

-27 + 40 + 30 = 0 \\

-10 + 36 + 10 = 0 \\

50 + 72 + 5 = 0

\end{cases}

$$

Evaluating, we get:

$$

\begin{cases}

43 = 0 \\

36 = 0 \\

127 = 0

\end{cases}

$$

Clearly, these equations are **not true**, so the answer you provided is **not correct**.

I hope this helps you understand how to solve linear equations. If you have any other questions, feel free to ask me.