Solve the system of equations.

$\left\{\begin{array}{l}3x-5y+4z=5\\ 5x+2y+z=0\\ 2x+3y-2z=3\end{array}\right.$

The system of equations is,

$\left\{\begin{array}{l}3x-5y+4z=5...........\left(1\right)\\ 5x+2y+z=0.............\left(2\right)\\ 2x+3y-2z=3...........\left(3\right)\end{array}\right.$

Eliminating $z$from equations (1) and (2).

$$\begin{gathered} 3x-5y+4z=5................\left(1\right) \\ -20x-8y-4z=0...........\left(2\right)\times -4 \end{gathered}$$

Solving the equations, we get,

$-17x-13y=5............\left(4\right)$

Eliminating $z$from the equations (2) and (3).

$$\begin{gathered} 10x+4y+2z=0........\left(2\right)\times 2 \\ 2x+3y-2z=3..........\left(3\right) \end{gathered}$$

Solving the equations, we get,

$12x+7y=3.......\left(5\right)$

Solving equations (4) and (5).

$$\begin{gathered} -119x-91y=35.........\left(4\right)\times 7 \\ 156x+91y=39...........\left(5\right)\times 13 \end{gathered}$$

Adding the equations, we get,

$$\begin{gathered} 37x=74 \\ x=2 \end{gathered}$$

Substituting $x=2$in the equation

$-17x-13y=5$,

$$\begin{gathered} -17\left(2\right)-13y=5 \\ -34-13y=5 \\ -13y=5+34 \\ -13y=39 \\ y=-3 \end{gathered}$$

Substituting $$\begin{gathered} x=2 \\ y=-3 \end{gathered}$$in the equation

$3x-5y+4z=5$,

$$\begin{gathered} 3\left(2\right)-5(-3)+4z=5 \\ 6+15+4z=5 \\ 4z=5-21 \\ 4z=-16 \\ z=-4 \end{gathered}$$

Substituting $$\begin{gathered} x=2 \\ y=-3 \\ z=-4 \end{gathered}$$in the equation

$$\begin{gathered} 3x-5y+4z=5 \\ 3\left(2\right)-5(-3)+4(-4)=5 \\ 6+15-16=5 \\ 5=5 \end{gathered}$$

This is true.

Substituting $$\begin{gathered} x=2 \\ y=-3 \\ z=-4 \end{gathered}$$in the equation

$5x+2y+z=0$,

$$\begin{gathered} 5\left(2\right)+2(-3)+(-4)=0 \\ 10-6-4=0 \\ 0=0 \end{gathered}$$

This is true.

Substituting $$\begin{gathered} x=2 \\ y=-3 \\ z=-4 \end{gathered}$$in the equation

$2x+3y-2z=3$

role="math" localid="1644599197913" $$\begin{gathered} 2\left(2\right)+3(-3)-2(-4)=3 \\ 4-9+8=3 \\ 3=3 \end{gathered}$$

This is true.