Solve {l}{2p+(1)q-3t=(4)}{(-1)p-q+(1)t=-3}{(-2)p-(-6)q-5t=(-7)}

$$\left\{ \begin{array} { l } { 2 p + ( 1 ) q - 3 t = ( 4 ) } \\ { ( - 1 ) p - q + ( 1 ) t = - 3 } \\ { ( - 2 ) p - ( - 6 ) q - 5 t = ( - 7 ) } \end{array} \right.$$

Solve for p, q, t

$$t = \frac{17}{15} = 1\frac{2}{15} \approx 1.133333333$$

$$p = \frac{49}{15} = 3\frac{4}{15} \approx 3.266666667$$

$$q=\frac{13}{15}\approx 0.866666667$$

Short Steps Using Substitution

Reorder the equations.

$$-p-q+1t=-3 2p+1q-3t=4 -2p-\left(-6q\right)-5t=-7$$

Solve -p-q+1t=-3 for p.

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

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

$$2\left(-q+t+3\right)+1q-3t=4 -2\left(-q+t+3\right)-\left(-6q\right)-5t=-7$$

Solve these equations for q and t respectively.

$$q=2-t t=\frac{8}{7}q+\frac{1}{7}$$

Substitute 2-t for q in the equation t=\frac{8}{7}q+\frac{1}{7}.

$$t=\frac{8}{7}\left(2-t\right)+\frac{1}{7}$$

Solve t=\frac{8}{7}\left(2-t\right)+\frac{1}{7} for t.

$$t=\frac{17}{15}$$

Substitute \frac{17}{15} for t in the equation q=2-t.

$$q=2-\frac{17}{15}$$

Calculate q from q=2-\frac{17}{15}.

$$q=\frac{13}{15}$$

Substitute \frac{13}{15} for q and \frac{17}{15} for t in the equation p=-q+t+3.

$$p=-\frac{13}{15}+\frac{17}{15}+3$$

Calculate p from p=-\frac{13}{15}+\frac{17}{15}+3.

$$p=\frac{49}{15}$$

The system is now solved.

$$p=\frac{49}{15} q=\frac{13}{15} t=\frac{17}{15}$$