$\left\{\begin{array} { l } 5x+3y-5z=45 \\ -5x-y+z=-29 \\ x=-\frac{ 3 }{ 4 }y-\frac{ 5 }{ 4 }z\end{array} \right.$
Substitute the given value of $x$ into the equation $5x+3y-5z=45$$\left\{\begin{array} { l } 5\left( -\frac{ 3 }{ 4 }y-\frac{ 5 }{ 4 }z \right)+3y-5z=45 \\ -5x-y+z=-29\end{array} \right.$
Substitute the given value of $x$ into the equation $-5x-y+z=-29$$\left\{\begin{array} { l } 5\left( -\frac{ 3 }{ 4 }y-\frac{ 5 }{ 4 }z \right)+3y-5z=45 \\ -5\left( -\frac{ 3 }{ 4 }y-\frac{ 5 }{ 4 }z \right)-y+z=-29\end{array} \right.$
Simplify the expression$\left\{\begin{array} { l } -3y-45z=180 \\ -5\left( -\frac{ 3 }{ 4 }y-\frac{ 5 }{ 4 }z \right)-y+z=-29\end{array} \right.$
Simplify the expression$\left\{\begin{array} { l } -3y-45z=180 \\ 11y+29z=-116\end{array} \right.$
Multiply both sides of the equation by $-11$$\left\{\begin{array} { l } 33y+495z=-1980 \\ 11y+29z=-116\end{array} \right.$
Multiply both sides of the equation by $-3$$\left\{\begin{array} { l } 33y+495z=-1980 \\ -33y-87z=348\end{array} \right.$
Sum the equations vertically to eliminate at least one variable$408z=-1632$
Divide both sides of the equation by $408$$z=-4$
Substitute the given value of $z$ into the equation $11y+29z=-116$$11y+29 \times \left( -4 \right)=-116$
Solve the equation for $y$$y=0$
Substitute the given values of $\begin{array} { l }y,& z\end{array}$ into the equation $x=-\frac{ 3 }{ 4 }y-\frac{ 5 }{ 4 }z$$x=-\frac{ 3 }{ 4 } \times 0-\frac{ 5 }{ 4 } \times \left( -4 \right)$
Simplify the expression$x=5$
The possible solution of the system is the ordered triple $\left( x, y, z\right)$$\left( x, y, z\right)=\left( 5, 0, -4\right)$
Check if the given ordered triple is a solution of the system of equations$\left\{\begin{array} { l } 5 \times 5+3 \times 0-5 \times \left( -4 \right)=45 \\ -5 \times 5-0+\left( -4 \right)=-29 \\ 4 \times 5+3 \times 0+5 \times \left( -4 \right)=0\end{array} \right.$
Simplify the equalities$\left\{\begin{array} { l } 45=45 \\ -29=-29 \\ 0=0\end{array} \right.$
Since all of the equalities are true, the ordered triple is the solution of the system$\left( x, y, z\right)=\left( 5, 0, -4\right)$