what is x y and z
4x-5y-z=-1
2x+2y-z=-4
3x+2y-2z=-4




To find the values of \( x \), \( y \), and \( z \), we can solve the system of linear equations:

\[
\begin{aligned}
1. & \quad 4x - 5y - z = -1 \\
2. & \quad 2x + 2y - z = -4 \\
3. & \quad 3x + 2y - 2z = -4
\end{aligned}
\]

We can solve this using either substitution, elimination, or matrix methods like Gaussian elimination. Let's use the elimination method to solve it step by step.

### Step 1: Eliminate one variable (say \( z \))

We can add or subtract equations to eliminate \( z \) from two pairs of equations.

- Subtract Equation (2) from Equation (1) to eliminate \( z \):

\[
(4x - 5y - z) - (2x + 2y - z) = (-1) - (-4)
\]

\[
(4x - 2x) + (-5y - 2y) + (-z + z) = 3
\]

\[
2x - 7y = 3 \quad \text{(Equation 4)}
\]

- Now subtract Equation (3) from Equation (2):

\[
(2x + 2y - z) - (3x + 2y - 2z) = (-4) - (-4)
\]

\[
(2x - 3x) + (2y - 2y) + (-z + 2z) = 0
\]

\[
-x + z = 0 \quad \text{(Equation 5)}
\]

### Step 2: Solve for \( z \) from Equation (5)

From Equation (5), we have:

\[
z = x
\]

### Step 3: Substitute \( z = x \) into the other equations

- Substitute \( z = x \) into Equation (2):

\[
2x + 2y - x = -4
\]

\[
x + 2y = -4 \quad \text{(Equation 6)}
\]

### Step 4: Solve the system with \( 2x - 7y = 3 \) and \( x + 2y = -4 \)

We now have two equations:

\[
1. \quad 2x - 7y = 3
\]
\[
2. \quad x + 2y = -4
\]

- Multiply Equation (2) by 2 to align the coefficients of \( x \):

\[
2x + 4y = -8
\]

- Subtract this from Equation (1):

\[
(2x - 7y) - (2x + 4y) = 3 - (-8)
\]

\[
-11y = 11
\]

\[
y = -1
\]

### Step 5: Substitute \( y = -1 \) into \( x + 2y = -4 \)

\[
x + 2(-1) = -4
\]

\[
x - 2 = -4
\]

\[
x = -2
\]

### Step 6: Find \( z \)

Since \( z = x \), we have:

\[
z = -2
\]

### Final Solution:

\[
x = -2, \quad y = -1, \quad z = -2
\]

Would you like more details on any step, or do you have any questions?

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Here are five related questions:
1. How do you solve systems of equations using the substitution method?
2. Can systems of equations be solved graphically?
3. What is Gaussian elimination, and how does it work?
4. How would the process change if we used matrices to solve the system?
5. Can all systems of equations be solved using elimination?

**Tip:** When solving a system of equations, choose the method (substitution, elimination, or matrices) based on the simplicity of the given equations.

What is x, y, and z for the system of equations: 4x - 5y - z = -1, 2x + 2y - z = -4, 3x + 2y - 2z = -4?

Find out how to solve a system of linear equations using elimination and substitution methods. This example covers solving for x, y, and z in a three-variable system with detailed step-by-step instructions. Ideal for students in Grades 9-12.

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