Let's solve the given system of equations step-by-step. Our system of equations is:
[tex]\[
\begin{cases}
x + y + z = 12 \quad \text{(Equation 1)} \\
2x - y + z = 7 \quad \text{(Equation 2)} \\
x + 2y - z = 6 \quad \text{(Equation 3)}
\end{cases}
\][/tex]
### Step 1: Eliminate one of the variables
First, let's eliminate [tex]\(z\)[/tex] by adding and subtracting the appropriate pairs of equations.
Add Equation 1 to Equation 2:
[tex]\[
(x + y + z) + (2x - y + z) = 12 + 7
\][/tex]
[tex]\[
3x + 2z = 19 \quad \text{(Equation 4)}
\][/tex]
Subtract Equation 3 from Equation 1:
[tex]\[
(x + y + z) - (x + 2y - z) = 12 - 6
\][/tex]
[tex]\[
-y + 2z = 6 \quad \text{(Equation 5)}
\][/tex]
### Step 2: Solve the new system of equations
Now we have a new system with two equations:
[tex]\[
\begin{cases}
3x + 2z = 19 \quad \text{(Equation 4)} \\
-y + 2z = 6 \quad \text{(Equation 5)}
\end{cases}
\][/tex]
Express [tex]\(y\)[/tex] from Equation 5:
[tex]\[
-y = 6 - 2z
\][/tex]
[tex]\[
y = 2z - 6 \quad \text{(Equation 6)}
\][/tex]
### Step 3: Substitute back to eliminate [tex]\(z\)[/tex]
Now substitute [tex]\(y\)[/tex] from Equation 6 into one of the original equations, let's use Equation 1:
[tex]\[
x + (2z - 6) + z = 12
\][/tex]
[tex]\[
x + 3z - 6 = 12
\][/tex]
[tex]\[
x + 3z = 18
\][/tex]
Now we substitute it into Equation 4 to solve for [tex]\(z\)[/tex]:
[tex]\[
3x + 2z = 19
\][/tex]
Using [tex]\(x = 18 - 3z\)[/tex], substitute into modified Equation 4:
[tex]\[
3(18 - 3z) + 2z = 19
\][/tex]
[tex]\[
54 - 9z + 2z = 19
\][/tex]
[tex]\[
54 - 7z = 19
\][/tex]
[tex]\[
-7z = -35
\][/tex]
[tex]\[
z = 5
\][/tex]
### Step 4: Solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]
Now we know [tex]\(z\)[/tex], substitute [tex]\(z = 5\)[/tex] back into the equation for [tex]\(x\)[/tex]:
[tex]\[
x + 3(5) = 18
\][/tex]
[tex]\[
x + 15 = 18
\][/tex]
[tex]\[
x = 3
\][/tex]
Substitute [tex]\(z = 5\)[/tex] back into Equation 6 to find [tex]\(y\)[/tex]:
[tex]\[
y = 2(5) - 6
\][/tex]
[tex]\[
y = 10 - 6
\][/tex]
[tex]\[
y = 4
\][/tex]
### Final answer
Hence, the solution to the system of equations is:
[tex]\[
x = 3, \quad y = 4, \quad z = 5
\][/tex]
So, [tex]\( (x, y, z) = (3.0, 4.0, 5.0) \)[/tex].
