# Question #ec492

Nov 21, 2016

The slope is $\frac{22}{89}$

#### Explanation:

Differentiate:

$8 x + 4 y + 4 x \frac{\mathrm{dy}}{\mathrm{dx}} + 6 {y}^{2} \frac{\mathrm{dy}}{\mathrm{dx}} = 0$

Move all of the terms without $\frac{\mathrm{dy}}{\mathrm{dx}}$ to the right:

$4 x \frac{\mathrm{dy}}{\mathrm{dx}} + 6 {y}^{2} \frac{\mathrm{dy}}{\mathrm{dx}} = - 8 x - 4 y$

Factor out $\frac{\mathrm{dy}}{\mathrm{dx}}$ on the left:

$\left(4 x + 6 {y}^{2}\right) \frac{\mathrm{dy}}{\mathrm{dx}} = - 8 x - 4 y$

Divide both sides by $\left(4 x + 6 {y}^{2}\right)$:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{- 8 x - 4 y}{4 x + 6 {y}^{2}}$

Remove a common factor of $\frac{2}{2}$:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{- 4 x - 2 y}{2 x + 3 {y}^{2}}$

The slope, m, is the above evaluated at $\left(- 7 , - 8\right)$

$m = \frac{- 4 \left(- 7\right) - 2 \left(- 8\right)}{2 \left(- 7\right) + 3 {\left(- 8\right)}^{2}}$

$m = \frac{44}{178} = \frac{22}{89}$