Solve {l}{3D-2E+F=-13}{6D+7E+F=-85}{5D+8E+F=-89}

$$\left. \begin{array} { l } { 3 D - 2 E + F = - 13 } \\ { 6 D + 7 E + F = - 85 } \\ { 5 D + 8 E + F = - 89 } \end{array} \right.$$

Solve for D, E, F

$$D=-3$$

$$E=-7$$

$$F=-18$$

Short Steps Using Substitution

Solve 3D-2E+F=-13 for F.

$$F=-3D+2E-13$$

Substitute -3D+2E-13 for F in the second and third equation.

$$6D+7E-3D+2E-13=-85 5D+8E-3D+2E-13=-89$$

Solve these equations for E and D respectively.

$$E=-8-\frac{1}{3}D D=-38-5E$$

Substitute -8-\frac{1}{3}D for E in the equation D=-38-5E.

$$D=-38-5\left(-8-\frac{1}{3}D\right)$$

Solve D=-38-5\left(-8-\frac{1}{3}D\right) for D.

$$D=-3$$

Substitute -3 for D in the equation E=-8-\frac{1}{3}D.

$$E=-8-\frac{1}{3}\left(-3\right)$$

Calculate E from E=-8-\frac{1}{3}\left(-3\right).

$$E=-7$$

Substitute -7 for E and -3 for D in the equation F=-3D+2E-13.

$$F=-3\left(-3\right)+2\left(-7\right)-13$$

Calculate F from F=-3\left(-3\right)+2\left(-7\right)-13.

$$F=-18$$

The system is now solved.

$$D=-3 E=-7 F=-18$$