To solve the given system of equations, we have:
[tex]\[
\begin{aligned}
1: & \quad 2x + 4y - 3z = 12 \\
2: & \quad -3x - 2y + 2z = -9 \\
3: & \quad -x + y - 3z = 0
\end{aligned}
\][/tex]
We need to find values for [tex]\( x \)[/tex], [tex]\( y \)[/tex], and [tex]\( z \)[/tex] that satisfy all three equations. Let's proceed step-by-step.
### Step 1: Solve for One Variable
Begin with equation [tex]\(3\)[/tex]:
[tex]\[ -x + y - 3z = 0 \][/tex]
Rearrange to solve for one variable, say [tex]\( y \)[/tex]:
[tex]\[ y = x + 3z \quad \text{(Equation 3')} \][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] into Equations 1 and 2
Substitute [tex]\( y = x + 3z \)[/tex] into equations 1 and 2:
#### For Equation 1:
[tex]\[
\begin{aligned}
2x + 4y - 3z &= 12 \\
2x + 4(x + 3z) - 3z &= 12 \\
2x + 4x + 12z - 3z &= 12 \\
6x + 9z &= 12 \\
\end{aligned}
\][/tex]
Simplify:
[tex]\[ 6x + 9z = 12 \quad \text{(Equation 1')} \][/tex]
Divide the entire equation by 3:
[tex]\[ 2x + 3z = 4 \quad \text{(Equation 1'')} \][/tex]
#### For Equation 2:
[tex]\[
\begin{aligned}
-3x - 2y + 2z &= -9 \\
-3x - 2(x + 3z) + 2z &= -9 \\
-3x - 2x - 6z + 2z &= -9 \\
-5x - 4z &= -9 \\
\end{aligned}
\][/tex]
Simplify:
[tex]\[ -5x - 4z = -9 \quad \text{(Equation 2')} \][/tex]
### Step 3: Solve the Simplified System of Two Variables
Now we have a simpler system of two equations with two variables:
[tex]\[
\begin{aligned}
1'': & \quad 2x + 3z = 4 \\
2': & \quad -5x - 4z = -9
\end{aligned}
\][/tex]
Let's solve these equations simultaneously.
#### Solve Equation 1'' for [tex]\( x \)[/tex]:
[tex]\[
2x + 3z = 4 \\
2x = 4 - 3z \\
x = \frac{4 - 3z}{2} \quad \text{(Equation 4)}
\][/tex]
#### Substitute Equation 4 into Equation 2':
[tex]\[
-5 \left( \frac{4 - 3z}{2} \right) - 4z = -9 \\
- \frac{20 - 15z}{2} - 4z = -9 \\
-10 + \frac{15z}{2} - 4z = -9 \\
-10 + \frac{15z}{2} - \frac{8z}{2} = -9 \\
-10 + \frac{7z}{2} = -9 \\
\frac{7z}{2} = 1 \\
7z = 2 \\
z = \frac{2}{7}
\][/tex]
### Step 4: Substitute [tex]\( z \)[/tex] into Equation 4 to Find [tex]\( x \)[/tex]:
[tex]\[
x = \frac{4 - 3z}{2} \\
x = \frac{4 - 3 \left( \frac{2}{7} \right)}{2} \\
x = \frac{4 - \frac{6}{7}}{2} \\
x = \frac{\frac{28}{7} - \frac{6}{7}}{2} \\
x = \frac{\frac{22}{7}}{2} \\
x = \frac{22}{14} \\
x = \frac{11}{7}
\][/tex]
### Step 5: Substitute [tex]\( x \)[/tex] and [tex]\( z \)[/tex] into Equation 3' to Find [tex]\( y \)[/tex]:
[tex]\[
y = x + 3z \\
y = \frac{11}{7} + 3 \left( \frac{2}{7} \right) \\
y = \frac{11}{7} + \frac{6}{7} \\
y = \frac{17}{7}
\][/tex]
### Final Solution:
The solutions to the system of equations are:
[tex]\[ x = \frac{11}{7}, \quad y = \frac{17}{7}, \quad z = \frac{2}{7} \][/tex]
These values satisfy all the given equations.
