Solve {c}{d-e-4t=-4}{2d+2e+f=11}{d+e+9f=13}

$$\left\{ \begin{array} { c } { d - e - 4 t = - 4 } \\ { 2 d + 2 e + f = 11 } \\ { d + e + 9 f = 13 } \end{array} \right.$$

Solve for d, t, f

$$t=-\frac{e}{2}+\frac{77}{34}\approx 0.905564968$$

$$d=\frac{86}{17}-e\approx 2.340541701$$

$$f=\frac{15}{17}\approx 0.882352941$$

Short Steps Using Substitution

Solve d-e-4t=-4 for d.

$$d=e+4t-4$$

Substitute e+4t-4 for d in the second and third equation.

$$2\left(e+4t-4\right)+2e+f=11 e+4t-4+e+9f=13$$

Solve these equations for t and f respectively.

$$t=\frac{19}{8}-\frac{1}{2}e-\frac{1}{8}f f=-\frac{2}{9}e+\frac{17}{9}-\frac{4}{9}t$$

Substitute \frac{19}{8}-\frac{1}{2}e-\frac{1}{8}f for t in the equation f=-\frac{2}{9}e+\frac{17}{9}-\frac{4}{9}t.

$$f=-\frac{2}{9}e+\frac{17}{9}-\frac{4}{9}\left(\frac{19}{8}-\frac{1}{2}e-\frac{1}{8}f\right)$$

Solve f=-\frac{2}{9}e+\frac{17}{9}-\frac{4}{9}\left(\frac{19}{8}-\frac{1}{2}e-\frac{1}{8}f\right) for f.

$$f=\frac{15}{17}$$

Substitute \frac{15}{17} for f in the equation t=\frac{19}{8}-\frac{1}{2}e-\frac{1}{8}f.

$$t=\frac{19}{8}-\frac{1}{2}e-\frac{1}{8}\times \frac{15}{17}$$

Calculate t from t=\frac{19}{8}-\frac{1}{2}e-\frac{1}{8}\times \frac{15}{17}.

$$t=-\frac{1}{2}e+\frac{77}{34}$$

Substitute -\frac{1}{2}e+\frac{77}{34} for t in the equation d=e+4t-4.

$$d=e+4\left(-\frac{1}{2}e+\frac{77}{34}\right)-4$$

Calculate d from d=e+4\left(-\frac{1}{2}e+\frac{77}{34}\right)-4.

$$d=-e+\frac{86}{17}$$

The system is now solved.

$$d=-e+\frac{86}{17} t=-\frac{1}{2}e+\frac{77}{34} f=\frac{15}{17}$$