https://file.aiquickdraw.com/mathbot/file/1726631632232ir2q8d6k.jpg

The system of equations from the image is:

\[
-4x + 6y + 3z = -10
\]
\[
4x - 10y - z = -10
\]
\[
-4x + y - z = -9
\]

Let's solve this system step-by-step using either substitution or elimination. I'll proceed with the elimination method.

### Step 1: Add the first and third equations
By adding these two equations, we aim to eliminate \( x \).

\[
(-4x + 6y + 3z) + (-4x + y - z) = -10 + (-9)
\]
Simplifying:

\[
-8x + 7y + 2z = -19
\]

This is our new equation (Equation 4):
\[
-8x + 7y + 2z = -19 \quad \text{(Equation 4)}
\]

### Step 2: Add the second and third equations
We also eliminate \( x \) from the second and third equations.

\[
(4x - 10y - z) + (-4x + y - z) = -10 + (-9)
\]
Simplifying:

\[
-9y - 2z = -19
\]
This is our new equation (Equation 5):
\[
-9y - 2z = -19 \quad \text{(Equation 5)}
\]

### Step 3: Solve for \( y \) and \( z \)
Now we solve the system formed by Equations 4 and 5:

From Equation 5:
\[
-9y - 2z = -19 \quad \text{(Equation 5)}
\]
Solve for \( z \):
\[
z = \frac{-19 + 9y}{-2} = \frac{19 - 9y}{2}
\]

### Step 4: Substitute \( z \) back into Equation 4 and solve for \( y \)
I will calculate the solution next.It seems I made a mistake in interpreting the structure of the system in terms of substitution.

I will recalculate the solution

Solve the following system of equations for all three variables: -4x + 6y + 3z = -10, 4x - 10y - z = -10, -4x + y - z = -9.

This page provides a step-by-step solution for solving a system of linear equations in three variables using the elimination method. Learn how to eliminate variables and solve for x, y, and z. Suitable for Grades 10-12.

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