Solve 3x^2+5x+y/4x+6y=5x+c

$$\frac { 3 x ^ { 2 } + 5 x + y } { 4 x + 6 y } = 5 x + c$$

Solve for c

$$c=\frac{y+5x-30xy-17x^{2}}{2\left(2x+3y\right)}$$

$$x\neq -\frac{3y}{2}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2\left(2x+3y\right).

$$3x^{2}+5x+y=5x\times 2\left(2x+3y\right)+2\left(2x+3y\right)c$$

Multiply 5 and 2 to get 10.

$$3x^{2}+5x+y=10x\left(2x+3y\right)+2\left(2x+3y\right)c$$

Use the distributive property to multiply 10x by 2x+3y.

$$3x^{2}+5x+y=20x^{2}+30xy+2\left(2x+3y\right)c$$

Use the distributive property to multiply 2 by 2x+3y.

$$3x^{2}+5x+y=20x^{2}+30xy+\left(4x+6y\right)c$$

Use the distributive property to multiply 4x+6y by c.

$$3x^{2}+5x+y=20x^{2}+30xy+4xc+6yc$$

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

$$20x^{2}+30xy+4xc+6yc=3x^{2}+5x+y$$

Subtract 20x^{2} from both sides.

$$30xy+4xc+6yc=3x^{2}+5x+y-20x^{2}$$

Combine 3x^{2} and -20x^{2} to get -17x^{2}.

$$30xy+4xc+6yc=-17x^{2}+5x+y$$

Subtract 30xy from both sides.

$$4xc+6yc=-17x^{2}+5x+y-30xy$$

Combine all terms containing c.

$$\left(4x+6y\right)c=-17x^{2}+5x+y-30xy$$

The equation is in standard form.

$$\left(4x+6y\right)c=y+5x-30xy-17x^{2}$$

Divide both sides by 4x+6y.

$$\frac{\left(4x+6y\right)c}{4x+6y}=\frac{y+5x-30xy-17x^{2}}{4x+6y}$$

Dividing by 4x+6y undoes the multiplication by 4x+6y.

$$c=\frac{y+5x-30xy-17x^{2}}{4x+6y}$$

Divide -17x^{2}+5x+y-30xy by 4x+6y.

$$c=\frac{y+5x-30xy-17x^{2}}{2\left(2x+3y\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{900y^{2}-168cy-232y+16c^{2}-40c+25}}{34}-\frac{2c}{17}-\frac{15y}{17}+\frac{5}{34}\text{, }&\left(y\neq \frac{26}{27}\text{ and }c\neq \frac{5}{4}\text{ and }arg(4c-5)\geq \pi \right)\text{ or }\left(arg(4c-5)\geq \pi \text{ and }c\neq \frac{5}{4}\text{ and }arg(4c-\frac{227}{9})\geq \pi \right)\text{ or }\left(c\neq \frac{227}{36}\text{ and }y\neq 0\text{ and }arg(4c-\frac{227}{9})\geq \pi \right)\text{ or }\left(y\neq \frac{26}{27}\text{ and }y\neq 0\right)\\x=-\frac{\sqrt{900y^{2}-168cy-232y+16c^{2}-40c+25}}{34}-\frac{2c}{17}-\frac{15y}{17}+\frac{5}{34}\text{, }&\left(y\neq \frac{26}{27}\text{ and }c\neq \frac{5}{4}\text{ and }arg(5-4c)\geq \pi \right)\text{ or }\left(arg(\frac{227}{9}-4c)\geq \pi \text{ and }c\neq \frac{227}{36}\text{ and }arg(5-4c)\geq \pi \right)\text{ or }\left(arg(\frac{227}{9}-4c)\geq \pi \text{ and }c\neq \frac{227}{36}\text{ and }y\neq 0\right)\text{ or }\left(y\neq \frac{26}{27}\text{ and }y\neq 0\right)\end{matrix}\right.$$