Answer
$\{\left(\frac{1}{2},3,-2 \right ) \}$.
Work Step by Step
The given equations are
$\Rightarrow 7z-3=2(x-3y)$
Apply distributive property
$\Rightarrow 7z-3=2x-6y$
Add $-2x+6y+3$ to both sides.
$\Rightarrow 7z-3-2x+6y+3=2x-6y-2x+6y+3$
Add like terms.
$\Rightarrow -2x+6y+7z=3$......(1)
$\Rightarrow 5y+3z-7=4x$
Add $-4x+7$ to both sides.
$\Rightarrow 5y+3z-7-4x+7=4x-4x+7$
Add like terms.
$\Rightarrow -4x+5y+3z=7$......(2)
$\Rightarrow 4+5z=3(2x-y)$
Apply distributive property
$\Rightarrow 4+5z=6x-3y$
Add $-6x+3y-4$ to both sides.
$\Rightarrow 4+5z-6x+3y-4=6x-3y-6x+3y-4$
Add like terms.
$\Rightarrow -6x+3y+5z=-4$.....(3)
Multiply equation (1) by $-2$.
$\Rightarrow 4x-12y-14z=-6$......(4)
Now add equations (2) and (4).
$\Rightarrow -4x+5y+3z+4x-12y-14z=7-6$
Add like terms.
$\Rightarrow -7y-11z=1$...... (5)
Multiply equation (1) by $-3$.
$\Rightarrow 6x-18y-21z=-9$......(6)
Now add equations (3) and (6).
$\Rightarrow -6x+3y+5z+6x-18y-21z=-4-9$
Add like terms.
$\Rightarrow -15y-16z=-13$......(7)
Multiply equation (5) by $-15$ and equation (7) by $7$ and then add.
$\Rightarrow -15(-7y-11z)+7(-15y-16z)=-15(1)+7(-13)$
Apply distributive property.
$\Rightarrow 105y+165z-105y-112z=-15-91$
Add like terms.
$\Rightarrow 53z=-106$
Divide both sides by $53$.
$\Rightarrow \frac{53z}{53}=\frac{-106}{53}$
Simplify.
$\Rightarrow z=-2$
Substitute the value of $z$ into equation (5).
$\Rightarrow -7y-11(-2)=1$
Simplify.
$\Rightarrow -7y+22=1$
Subtract $22$ from both sides.
$\Rightarrow -7y+22-22=1-22$
Simplify.
$\Rightarrow -7y=-21$
Divide both sides by $-7$.
$\Rightarrow y=3$
Substitute the values of $y$ and $z$ into equation (1).
$\Rightarrow -2x+6(3)+7(-2)=3$
Simplify.
$\Rightarrow -2x+18-14=3$
$\Rightarrow -2x+4=3$
Subtract $4$ from both sides.
$\Rightarrow -2x+4-4=3-4$
$\Rightarrow -2x=-1$
Divide both sides by $-2$.
$\Rightarrow x=\frac{1}{2}$
The solution set is $\{(x,y,z)\}=\{\left(\frac{1}{2},3,-2 \right ) \}$.
