Solve {l}{2g+2f+c=-2}{4g-2f+c=-5}{6g+4f+c=-13}

$$\left. \begin{array} { l } { 2 g + 2 f + c = - 2 } \\ { 4 g - 2 f + c = - 5 } \\ { 6 g + 4 f + c = - 13 } \end{array} \right.$$

Solve for g, f, c

$$g = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

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

$$c=4$$

Short Steps Using Substitution

Solve 2g+2f+c=-2 for c.

$$c=-2-2g-2f$$

Substitute -2-2g-2f for c in the second and third equation.

$$4g-2f-2-2g-2f=-5 6g+4f-2-2g-2f=-13$$

Solve these equations for f and g respectively.

$$f=\frac{1}{2}g+\frac{3}{4} g=-\frac{1}{2}f-\frac{11}{4}$$

Substitute \frac{1}{2}g+\frac{3}{4} for f in the equation g=-\frac{1}{2}f-\frac{11}{4}.

$$g=-\frac{1}{2}\left(\frac{1}{2}g+\frac{3}{4}\right)-\frac{11}{4}$$

Solve g=-\frac{1}{2}\left(\frac{1}{2}g+\frac{3}{4}\right)-\frac{11}{4} for g.

$$g=-\frac{5}{2}$$

Substitute -\frac{5}{2} for g in the equation f=\frac{1}{2}g+\frac{3}{4}.

$$f=\frac{1}{2}\left(-\frac{5}{2}\right)+\frac{3}{4}$$

Calculate f from f=\frac{1}{2}\left(-\frac{5}{2}\right)+\frac{3}{4}.

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

Substitute -\frac{1}{2} for f and -\frac{5}{2} for g in the equation c=-2-2g-2f.

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

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

$$c=4$$

The system is now solved.

$$g=-\frac{5}{2} f=-\frac{1}{2} c=4$$