Solve F=x+3y+lambda(7.5-xy)

$$F = x + 3 y + \lambda ( 7.5 - x y )$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{6y+15\lambda -2F}{2\left(1-y\lambda \right)}\text{, }&\lambda =0\text{ or }y\neq \frac{1}{\lambda }\\x\in \mathrm{C}\text{, }&F=\frac{15\lambda }{2}+\frac{3}{\lambda }\text{ and }y=\frac{1}{\lambda }\text{ and }\lambda \neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \lambda by 7.5-xy.

$$F=x+3y+7.5\lambda -\lambda xy$$

Swap sides so that all variable terms are on the left hand side.

$$x+3y+7.5\lambda -\lambda xy=F$$

Subtract 3y from both sides.

$$x+7.5\lambda -\lambda xy=F-3y$$

Subtract 7.5\lambda from both sides.

$$x-\lambda xy=F-3y-7.5\lambda$$

Combine all terms containing x.

$$\left(1-\lambda y\right)x=F-3y-7.5\lambda$$

The equation is in standard form.

$$\left(1-y\lambda \right)x=-\frac{15\lambda }{2}+F-3y$$

Divide both sides by 1-\lambda y.

$$\frac{\left(1-y\lambda \right)x}{1-y\lambda }=\frac{-\frac{15\lambda }{2}+F-3y}{1-y\lambda }$$

Dividing by 1-\lambda y undoes the multiplication by 1-\lambda y.

$$x=\frac{-\frac{15\lambda }{2}+F-3y}{1-y\lambda }$$

Divide F-3y-\frac{15\lambda }{2} by 1-\lambda y.

$$x=\frac{-6y+2F-15\lambda }{2\left(1-y\lambda \right)}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{6y+15\lambda -2F}{2\left(1-y\lambda \right)}\text{, }&\lambda =0\text{ or }y\neq \frac{1}{\lambda }\\x\in \mathrm{R}\text{, }&F=\frac{15\lambda }{2}+\frac{3}{\lambda }\text{ and }y=\frac{1}{\lambda }\text{ and }\lambda \neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply \lambda by 7.5-xy.

$$F=x+3y+7.5\lambda -\lambda xy$$

Swap sides so that all variable terms are on the left hand side.

$$x+3y+7.5\lambda -\lambda xy=F$$

Subtract 3y from both sides.

$$x+7.5\lambda -\lambda xy=F-3y$$

Subtract 7.5\lambda from both sides.

$$x-\lambda xy=F-3y-7.5\lambda$$

Combine all terms containing x.

$$\left(1-\lambda y\right)x=F-3y-7.5\lambda$$

The equation is in standard form.

$$\left(1-y\lambda \right)x=-\frac{15\lambda }{2}+F-3y$$

Divide both sides by 1-\lambda y.

$$\frac{\left(1-y\lambda \right)x}{1-y\lambda }=\frac{-\frac{15\lambda }{2}+F-3y}{1-y\lambda }$$

Dividing by 1-\lambda y undoes the multiplication by 1-\lambda y.

$$x=\frac{-\frac{15\lambda }{2}+F-3y}{1-y\lambda }$$

Divide F-3y-\frac{15\lambda }{2} by 1-\lambda y.

$$x=\frac{-6y+2F-15\lambda }{2\left(1-y\lambda \right)}$$

Solve for F

$$F=-xy\lambda +\frac{15\lambda }{2}+x+3y$$