Solve {c}{s+2(q-p)=5}{3q+2s=-6p}{4q+p=s/3}

$$\left. \begin{array} { c } { s + 2 ( q - p ) = 5 } \\ { 3 q + 2 s = - 6 p } \\ { 4 q + p = \frac { s } { 3 } } \end{array} \right.$$

Solve for s, q, p

$$s = \frac{105}{47} = 2\frac{11}{47} \approx 2.234042553$$

$$q=\frac{20}{47}\approx 0.425531915$$

$$p=-\frac{45}{47}\approx -0.957446809$$

Short Steps Using Substitution

Multiply each equation by the least common multiple of denominators in it. Simplify.

$$s+2\left(q-p\right)=5 3q+2s=-6p 12q+3p=s$$

Reorder the equations.

$$12q+3p=s 3q+2s=-6p s+2\left(q-p\right)=5$$

Solve 12q+3p=s for s.

$$s=12q+3p$$

Substitute 12q+3p for s in the second and third equation.

$$3q+2\left(12q+3p\right)=-6p 12q+3p+2\left(q-p\right)=5$$

Solve these equations for q and p respectively.

$$q=-\frac{4}{9}p p=-14q+5$$

Substitute -\frac{4}{9}p for q in the equation p=-14q+5.

$$p=-14\left(-\frac{4}{9}\right)p+5$$

Solve p=-14\left(-\frac{4}{9}\right)p+5 for p.

$$p=-\frac{45}{47}$$

Substitute -\frac{45}{47} for p in the equation q=-\frac{4}{9}p.

$$q=-\frac{4}{9}\left(-\frac{45}{47}\right)$$

Calculate q from q=-\frac{4}{9}\left(-\frac{45}{47}\right).

$$q=\frac{20}{47}$$

Substitute \frac{20}{47} for q and -\frac{45}{47} for p in the equation s=12q+3p.

$$s=12\times \frac{20}{47}+3\left(-\frac{45}{47}\right)$$

Calculate s from s=12\times \frac{20}{47}+3\left(-\frac{45}{47}\right).

$$s=\frac{105}{47}$$

The system is now solved.

$$s=\frac{105}{47} q=\frac{20}{47} p=-\frac{45}{47}$$