Solve {l}{2p+(-3)q-3t=(3)}{(-5)p-q+(3)t=-3}{(4)p-(0)q-5t=(-8)}

$$\left\{ \begin{array} { l } { 2 p + ( - 3 ) q - 3 t = ( 3 ) } \\ { ( - 5 ) p - q + ( 3 ) t = - 3 } \\ { ( 4 ) p - ( 0 ) q - 5 t = ( - 8 ) } \end{array} \right.$$

Solve for p, q, t

$$t = \frac{184}{37} = 4\frac{36}{37} \approx 4.972972973$$

$$p = \frac{156}{37} = 4\frac{8}{37} \approx 4.216216216$$

$$q = -\frac{117}{37} = -3\frac{6}{37} \approx -3.162162162$$

Short Steps Using Substitution

Reorder the equations.

$$-5p-q+3t=-3 2p-3q-3t=3 4p-0q-5t=-8$$

Solve -5p-q+3t=-3 for q.

$$q=-5p+3t+3$$

Substitute -5p+3t+3 for q in the second and third equation.

$$2p-3\left(-5p+3t+3\right)-3t=3 4p-0\left(-5p+3t+3\right)-5t=-8$$

Solve these equations for p and t respectively.

$$p=\frac{12}{17}+\frac{12}{17}t t=\frac{8}{5}+\frac{4}{5}p$$

Substitute \frac{12}{17}+\frac{12}{17}t for p in the equation t=\frac{8}{5}+\frac{4}{5}p.

$$t=\frac{8}{5}+\frac{4}{5}\left(\frac{12}{17}+\frac{12}{17}t\right)$$

Solve t=\frac{8}{5}+\frac{4}{5}\left(\frac{12}{17}+\frac{12}{17}t\right) for t.

$$t=\frac{184}{37}$$

Substitute \frac{184}{37} for t in the equation p=\frac{12}{17}+\frac{12}{17}t.

$$p=\frac{12}{17}+\frac{12}{17}\times \frac{184}{37}$$

Calculate p from p=\frac{12}{17}+\frac{12}{17}\times \frac{184}{37}.

$$p=\frac{156}{37}$$

Substitute \frac{156}{37} for p and \frac{184}{37} for t in the equation q=-5p+3t+3.

$$q=-5\times \frac{156}{37}+3\times \frac{184}{37}+3$$

Calculate q from q=-5\times \frac{156}{37}+3\times \frac{184}{37}+3.

$$q=-\frac{117}{37}$$

The system is now solved.

$$p=\frac{156}{37} q=-\frac{117}{37} t=\frac{184}{37}$$