To solve the system of equations:
[tex]\[
\begin{align*}
1. & \quad x + y - z = 4 \\
2. & \quad 2x + z = 7 \\
3. & \quad 3x - 2y = 5
\end{align*}
\][/tex]
### Step 1: Solve Equation 2 for [tex]\( z \)[/tex]
From equation [tex]\( 2x + z = 7 \)[/tex]:
[tex]\[
z = 7 - 2x
\][/tex]
### Step 2: Substitute [tex]\( z \)[/tex] into Equation 1
Substitute [tex]\( z = 7 - 2x \)[/tex] into equation [tex]\( x + y - z = 4 \)[/tex]:
[tex]\[
x + y - (7 - 2x) = 4
\][/tex]
Simplifying this:
[tex]\[
x + y - 7 + 2x = 4 \\
3x + y - 7 = 4 \\
3x + y = 11 \quad \text{(Equation 4)}
\][/tex]
### Step 3: Solve for [tex]\( y \)[/tex] using Equation 4
From equation [tex]\( 3x + y = 11 \)[/tex]:
[tex]\[
y = 11 - 3x
\][/tex]
### Step 4: Substitute [tex]\( y \)[/tex] into Equation 3
Substitute [tex]\( y = 11 - 3x \)[/tex] into equation [tex]\( 3x - 2y = 5 \)[/tex]:
[tex]\[
3x - 2(11 - 3x) = 5
\][/tex]
Simplifying this:
[tex]\[
3x - 22 + 6x = 5 \\
9x - 22 = 5 \\
9x = 27 \\
x = 3
\][/tex]
### Step 5: Substitute [tex]\( x \)[/tex] back to find [tex]\( y \)[/tex] and [tex]\( z \)[/tex]
Substitute [tex]\( x = 3 \)[/tex] into [tex]\( y = 11 - 3x \)[/tex]:
[tex]\[
y = 11 - 3(3) \\
y = 11 - 9 \\
y = 2
\][/tex]
Substitute [tex]\( x = 3 \)[/tex] into [tex]\( z = 7 - 2x \)[/tex]:
[tex]\[
z = 7 - 2(3) \\
z = 7 - 6 \\
z = 1
\][/tex]
### Summary
The solution to the system of equations is:
[tex]\[
(x, y, z) = (3, 2, 1)
\][/tex]
### Verification
To ensure the solution is correct, we can substitute [tex]\( (x, y, z) = (3, 2, 1) \)[/tex] back into the original equations:
[tex]\[
1. \quad 3 + 2 - 1 = 4 \quad \text{(True)} \\
2. \quad 2(3) + 1 = 7 \quad \text{(True)} \\
3. \quad 3(3) - 2(2) = 5 \quad \text{(True)}
\][/tex]
Since all three equations hold true, [tex]\((x, y, z) = (3, 2, 1)\)[/tex] is indeed the correct solution.
