Solve {l}{a-p+q=0}{4a-2p+q=5}{9a+3p+q=-15}

$$\left\{ \begin{array} { l } { a - p + q = 0 } \\ { 4 a - 2 p + q = 5 } \\ { 9 a + 3 p + q = - 15 } \end{array} \right.$$

Solve for a, p, q

$$a=\frac{1}{4}=0.25$$

$$p = -\frac{17}{4} = -4\frac{1}{4} = -4.25$$

$$q = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

Short Steps Using Substitution

Solve a-p+q=0 for a.

$$a=p-q$$

Substitute p-q for a in the second and third equation.

$$4\left(p-q\right)-2p+q=5 9\left(p-q\right)+3p+q=-15$$

Solve these equations for p and q respectively.

$$p=\frac{3}{2}q+\frac{5}{2} q=\frac{15}{8}+\frac{3}{2}p$$

Substitute \frac{3}{2}q+\frac{5}{2} for p in the equation q=\frac{15}{8}+\frac{3}{2}p.

$$q=\frac{15}{8}+\frac{3}{2}\left(\frac{3}{2}q+\frac{5}{2}\right)$$

Solve q=\frac{15}{8}+\frac{3}{2}\left(\frac{3}{2}q+\frac{5}{2}\right) for q.

$$q=-\frac{9}{2}$$

Substitute -\frac{9}{2} for q in the equation p=\frac{3}{2}q+\frac{5}{2}.

$$p=\frac{3}{2}\left(-\frac{9}{2}\right)+\frac{5}{2}$$

Calculate p from p=\frac{3}{2}\left(-\frac{9}{2}\right)+\frac{5}{2}.

$$p=-\frac{17}{4}$$

Substitute -\frac{17}{4} for p and -\frac{9}{2} for q in the equation a=p-q.

$$a=-\frac{17}{4}-\left(-\frac{9}{2}\right)$$

Calculate a from a=-\frac{17}{4}-\left(-\frac{9}{2}\right).

$$a=\frac{1}{4}$$

The system is now solved.

$$a=\frac{1}{4} p=-\frac{17}{4} q=-\frac{9}{2}$$