−2x − 5y + 4z = 21
−5x − 5y + z = 21
−4y − 4z = 8

You have a system of three linear equations:

1. \( -2x - 5y + 4z = 21 \)
2. \( -5x - 5y + z = 21 \)
3. \( -4y - 4z = 8 \)

Let's solve this system step by step using the substitution or elimination method.

### Step 1: Simplify the third equation
The third equation can be simplified by dividing the entire equation by \(-4\):

\[
-4y - 4z = 8 \quad \Rightarrow \quad y + z = -2
\]

So, the simplified equation is:

\[
y + z = -2 \quad \text{(Equation 4)}
\]

### Step 2: Solve for \(z\) in terms of \(y\) from Equation 4
\[
z = -2 - y \quad \text{(Equation 5)}
\]

### Step 3: Substitute \(z = -2 - y\) into the other two equations

#### Substitute into Equation 1:
\[
-2x - 5y + 4(-2 - y) = 21
\]
Simplifying:
\[
-2x - 5y - 8 - 4y = 21
\]
\[
-2x - 9y = 29 \quad \text{(Equation 6)}
\]

#### Substitute into Equation 2:
\[
-5x - 5y + (-2 - y) = 21
\]
Simplifying:
\[
-5x - 5y - 2 - y = 21
\]
\[
-5x - 6y = 23 \quad \text{(Equation 7)}
\]

### Step 4: Solve the system of two equations (Equations 6 and 7)

We now have the system:
1. \( -2x - 9y = 29 \)
2. \( -5x - 6y = 23 \)

Let's solve this system using the elimination method. First, multiply Equation 6 by 5 and Equation 7 by 2 to align the \(x\)-terms:

\[
5(-2x - 9y) = 5(29) \quad \Rightarrow \quad -10x - 45y = 145
\]
\[
2(-5x - 6y) = 2(23) \quad \Rightarrow \quad -10x - 12y = 46
\]

### Step 5: Subtract the two equations
\[
(-10x - 45y) - (-10x - 12y) = 145 - 46
\]
Simplifying:
\[
-33y = 99
\]
\[
y = -3
\]

### Step 6: Substitute \(y = -3\) into Equation 5
Using \(y = -3\) in \(z = -2 - y\):
\[
z = -2 - (-3) = 1
\]

### Step 7: Substitute \(y = -3\) and \(z = 1\) into one of the original equations, say Equation 1, to find \(x\):
\[
-2x - 5(-3) + 4(1) = 21
\]
Simplifying:
\[
-2x + 15 + 4 = 21
\]
\[
-2x + 19 = 21
\]
\[
-2x = 2
\]
\[
x = -1
\]

### Final Solution:
The solution to the system is:
\[
x = -1, \quad y = -3, \quad z = 1
\]

Would you like further clarification on any step?

Here are some related questions to consider:
1. How would you approach solving a system with more variables?
2. Can systems of equations have no solutions or infinite solutions?
3. How can you verify that the solution is correct?
4. What is the difference between substitution and elimination methods?
5. Can systems of equations be solved graphically?

**Tip:** Always check your solution by substituting the values back into the original equations.

Learn how to solve the system of equations −2x − 5y + 4z = 21, −5x − 5y + z = 21, and −4y − 4z = 8 using the elimination and substitution methods. This problem covers key algebraic concepts like solving systems of linear equations with three variables. Suitable for students in Grades 9-12.

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