Given an equation {2x+3y-z= 9, 4x-5y+2z=1, x+y+z=9} The z value is...?

We can solve this system of equations using substitution or elimination methods to find the value of \( z \).

The system of equations is:

\[
\begin{aligned}
1. & \ 2x + 3y - z = 9 \\
2. & \ 4x - 5y + 2z = 1 \\
3. & \ x + y + z = 9
\end{aligned}
\]

### Step 1: Solve Equation 3 for \( z \)
From the third equation \( x + y + z = 9 \), solve for \( z \):
\[
z = 9 - x - y
\]

### Step 2: Substitute \( z = 9 - x - y \) into Equations 1 and 2
Substitute this expression for \( z \) into the first two equations.

#### For Equation 1:
\[
2x + 3y - (9 - x - y) = 9
\]
Simplify:
\[
2x + 3y - 9 + x + y = 9
\]
\[
3x + 4y - 9 = 9
\]
\[
3x + 4y = 18 \quad \text{(Equation 4)}
\]

#### For Equation 2:
\[
4x - 5y + 2(9 - x - y) = 1
\]
Simplify:
\[
4x - 5y + 18 - 2x - 2y = 1
\]
\[
2x - 7y + 18 = 1
\]
\[
2x - 7y = -17 \quad \text{(Equation 5)}
\]

### Step 3: Solve the system of Equations 4 and 5
Now, we solve the system:
\[
\begin{aligned}
3x + 4y &= 18 \quad \text{(Equation 4)} \\
2x - 7y &= -17 \quad \text{(Equation 5)}
\end{aligned}
\]

#### Multiply Equation 4 by 2 and Equation 5 by 3 to eliminate \( x \):
\[
6x + 8y = 36
\]
\[
6x - 21y = -51
\]

### Step 4: Subtract the second equation from the first:
\[
(6x + 8y) - (6x - 21y) = 36 - (-51)
\]
\[
29y = 87
\]
\[
y = 3
\]

### Step 5: Substitute \( y = 3 \) back into Equation 4 to solve for \( x \):
\[
3x + 4(3) = 18
\]
\[
3x + 12 = 18
\]
\[
3x = 6
\]
\[
x = 2
\]

### Step 6: Substitute \( x = 2 \) and \( y = 3 \) into the expression for \( z \):
\[
z = 9 - x - y = 9 - 2 - 3 = 4
\]

Thus, the value of \( z \) is \( \boxed{4} \).

Would you like further details on any part of the solution?

Here are some related questions:
1. How do you solve systems of equations using the substitution method?
2. What is the elimination method for solving systems of equations?
3. Can systems of equations have no solution or infinitely many solutions?
4. How do you verify the solution of a system of equations?
5. How are systems of linear equations used in real-world applications?

**Tip**: When solving systems of equations, it's helpful to simplify and organize the process step by step, either by eliminating variables or substituting known expressions.

Learn how to solve a system of linear equations: 2x+3y-z=9, 4x-5y+2z=1, and x+y+z=9. This step-by-step solution covers key concepts such as the substitution and elimination methods, with detailed formulas and explanations. Ideal for students in Grades 9-12.

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