https://file.aiquickdraw.com/mathbot/file/17272743705187duc6da8.jpg

The system of equations provided in the image is:

\[
\begin{aligned}
-2x + 3y + 5z &= -21 \quad \text{(Equation 1)} \\
-4z &= 20 \quad \text{(Equation 2)} \\
6x - 3y &= 0 \quad \text{(Equation 3)}
\end{aligned}
\]

We are asked to find the value of \(z\).

### Step 1: Solve for \(z\) from Equation 2
From Equation 2, we have:

\[
-4z = 20
\]

Dividing both sides by \(-4\):

\[
z = \frac{20}{-4} = -5
\]

So, \(z = -5\).

### Step 2: Check if you want further details
Would you like further details on solving the entire system, such as calculating \(x\) and \(y\), or just solving for \(z\) is sufficient?

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**Here are 5 related questions:**
1. How do we solve for \(x\) and \(y\) in this system of equations?
2. What is the geometrical interpretation of a system of three linear equations?
3. How can we verify the solution by substituting into all equations?
4. What happens if the system has no solution or infinitely many solutions?
5. How do we solve systems of equations using matrix methods like Gaussian elimination?

**Tip**: Always check the consistency of your system after solving for one variable to ensure there are no contradictions in the equations!

What is the value of z in the system of equations?

-2x + 3y + 5z = -21
-4z = 20
6x - 3y = 0

Learn how to solve a linear system of equations with three variables to find the value of z. This step-by-step solution includes solving by substitution, isolating variables, and applying algebraic techniques. Suitable for Grades 8-10.

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