Answer
$\{(-2,0,4)\}$.
Work Step by Step
We will solve the system of equations using the Elimination Method. Let's note the equations:
$\left\{\begin{matrix}
4x& +3y &+3z&=&4& ...... (1) \\
3x& & +2z&=&2& ...... (2)\\
2x&-5y & &=&-4& ...... (3)
\end{matrix}\right.$
Step 1:- Reduce the system to two equations in two variables.
Multiply the equation (1) by $-2$.
$\Rightarrow -8x-6y -6z=-8 $ ...... (4)
Multiply the equation (2) by $3$.
$\Rightarrow 9x+ 6z=6 $ ...... (5)
Add equation (4) and (5).
$\Rightarrow -8x-6y-6z+9x+6z=-8+6 $
Simplify.
$\Rightarrow x-6y =-2 $
Multiply both sides by $-2$
$\Rightarrow -2x+12y =4 $ ...... (6)
Step 2:- Solve the two equations from the step 1.
Add equation (3) and (6).
$\Rightarrow 2x-5y-2x+12y =-4+4 $
Add like terms.
$\Rightarrow 7y =0$
Divide both sides by $7$.
$\Rightarrow \frac{7y}{7} =\frac{0}{7}$
Simplify.
$\Rightarrow y =0$
Step 3:- Use back-substitution to find the second variable.
Substitute the value of $y$ into equation (3).
$\Rightarrow 2x -5(0) =-4$
Simplify.
$\Rightarrow 2x =-4$
Divide both sides by $2$.
$\Rightarrow \frac{2x}{2} =\frac{-4}{2}$
Simplify.
$\Rightarrow x =-2$
Step 4:- Use back-substitution to find the remaining variable.
Substitute the value of $x$ into equation (2)
$\Rightarrow 3(-2) +2z =2$
Simplify.
$\Rightarrow -6 +2z =2$
Add $6$ to both sides.
$\Rightarrow -6 +2z+6 =2+6$
Simplify.
$\Rightarrow 2z =8$
Divide both sides by $2$.
$\Rightarrow \frac{2z}{2} =\frac{8}{2}$
Simplify.
$\Rightarrow z =4$
The solution set is $\{(x,y,z)\}=\{(-2,0,4)\}$.
