Answer
$\{\left ( 4,-2,3\right )\}$.
Work Step by Step
The given system of equations is
$\Rightarrow x-3z= -5$...... (1)
$\Rightarrow 2x-y+2z= 16 $...... (2)
$\Rightarrow 7x-3y-5z= 19$...... (3)
Multiply equation (2) by $-3$ and add to equation (3).
$\Rightarrow -3(2x-y+2z)+7x-3y-5z= -3(16)+19$
Apply distributive property.
$\Rightarrow -6x+3y-6z+7x-3y-5z= -48+19$
Simplify.
$\Rightarrow x-11z= -29$
Multiply the equation by $-1$
$\Rightarrow -1(x-11z)=-1( -29)$
$\Rightarrow -x+11z=29$...... (4).
Add equation (1) and equation (4).
$\Rightarrow x-3z-x+11z= -5+29$
Simplify.
$\Rightarrow 8z= 24$
Divide both sides by $8$.
$\Rightarrow \frac{8z}{8}= \frac{24}{8}$
Simplify.
$\Rightarrow z=3$
Substitute the value of $z$ into equation (1).
$\Rightarrow x-3(3)= -5$
Simplify.
$\Rightarrow x-9= -5$
Add $9$ to both sides.
$\Rightarrow x-9+9= -5+9$
Simplify.
$\Rightarrow x= 4$
Substitute the values of $x$ and $z$ into equation (2).
$\Rightarrow 2(4)-y+2(3)= 16 $
Simplify.
$\Rightarrow 8-y+6= 16 $
$\Rightarrow -y+14= 16 $
Subtract $14$ from both sides.
$\Rightarrow -y+14-14= 16-14 $
Simplify.
$\Rightarrow -y= 2 $
$\Rightarrow y= -2 $
The solution set is $\{\left ( x,y,z\right )\}=\{\left ( 4,-2,3\right )\}$.
