Calculate: {\text{begin}array l 5x+3y-5z=45 }-5x-y+z=-29 4x+3y+5z=0\text{end}array .

Expression: $\left\{\begin{array} { l } 5x+3y-5z=45 \\ -5x-y+z=-29 \\ 4x+3y+5z=0\end{array} \right.$

Solve the equation for $x$

$\left\{\begin{array} { l } 5x+3y-5z=45 \\ -5x-y+z=-29 \\ x=-\frac{ 3 }{ 4 }y-\frac{ 5 }{ 4 }z\end{array} \right.$

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

$\left( x, y, z\right)=\left( 5, 0, -4\right)$