Solve {l}{16p+4q+r=17}{4p+2q+r=7}{36p+6q+r=23}

$$\left. \begin{array} { l } { 16 p + 4 q + r = 17 } \\ { 4 p + 2 q + r = 7 } \\ { 36 p + 6 q + r = 23 } \end{array} \right.$$

Solve for p, q, r

$$r=-7$$

$$p=-\frac{1}{2}=-0.5$$

$$q=8$$

Short Steps Using Substitution

Solve 16p+4q+r=17 for r.

$$r=-16p-4q+17$$

Substitute -16p-4q+17 for r in the second and third equation.

$$4p+2q-16p-4q+17=7 36p+6q-16p-4q+17=23$$

Solve these equations for q and p respectively.

$$q=5-6p p=\frac{3}{10}-\frac{1}{10}q$$

Substitute 5-6p for q in the equation p=\frac{3}{10}-\frac{1}{10}q.

$$p=\frac{3}{10}-\frac{1}{10}\left(5-6p\right)$$

Solve p=\frac{3}{10}-\frac{1}{10}\left(5-6p\right) for p.

$$p=-\frac{1}{2}$$

Substitute -\frac{1}{2} for p in the equation q=5-6p.

$$q=5-6\left(-\frac{1}{2}\right)$$

Calculate q from q=5-6\left(-\frac{1}{2}\right).

$$q=8$$

Substitute 8 for q and -\frac{1}{2} for p in the equation r=-16p-4q+17.

$$r=-16\left(-\frac{1}{2}\right)-4\times 8+17$$

Calculate r from r=-16\left(-\frac{1}{2}\right)-4\times 8+17.

$$r=-7$$

The system is now solved.

$$p=-\frac{1}{2} q=8 r=-7$$