Solve {c}{8p+4q-3r=6}{p+3q-r=7}{4r-8=5q-4p}

$$\left\{ \begin{array} { c } { 8 p + 4 q - 3 r = 6 } \\ { p + 3 q - r = 7 } \\ { 4 r - 8 = 5 q - 4 p } \end{array} \right.$$

Solve for p, q, r

$$r=6$$

$$p=1$$

$$q=4$$

Short Steps Using Substitution

Reorder the equations.

$$p+3q-r=7 8p+4q-3r=6 4r-8=5q-4p$$

Solve p+3q-r=7 for p.

$$p=-3q+r+7$$

Substitute -3q+r+7 for p in the second and third equation.

$$8\left(-3q+r+7\right)+4q-3r=6 4r-8=5q-4\left(-3q+r+7\right)$$

Solve these equations for q and r respectively.

$$q=\frac{5}{2}+\frac{1}{4}r r=-\frac{5}{2}+\frac{17}{8}q$$

Substitute \frac{5}{2}+\frac{1}{4}r for q in the equation r=-\frac{5}{2}+\frac{17}{8}q.

$$r=-\frac{5}{2}+\frac{17}{8}\left(\frac{5}{2}+\frac{1}{4}r\right)$$

Solve r=-\frac{5}{2}+\frac{17}{8}\left(\frac{5}{2}+\frac{1}{4}r\right) for r.

$$r=6$$

Substitute 6 for r in the equation q=\frac{5}{2}+\frac{1}{4}r.

$$q=\frac{5}{2}+\frac{1}{4}\times 6$$

Calculate q from q=\frac{5}{2}+\frac{1}{4}\times 6.

$$q=4$$

Substitute 4 for q and 6 for r in the equation p=-3q+r+7.

$$p=-3\times 4+6+7$$

Calculate p from p=-3\times 4+6+7.

$$p=1$$

The system is now solved.

$$p=1 q=4 r=6$$