Solve 5x-4y/x+5=5y-14/y+3

$$\frac { 5 x - 4 y } { x + 5 } = \frac { 5 y - 14 } { y + 3 }$$

Solve for x

$$x=\frac{4y^{2}+37y-70}{29}$$

$$y\neq -\frac{25}{4}\text{ and }y\neq -3$$

Steps for Solving Linear Equation

Variable x cannot be equal to -5 since division by zero is not defined. Multiply both sides of the equation by \left(x+5\right)\left(y+3\right), the least common multiple of x+5,y+3.

$$\left(y+3\right)\left(5x-4y\right)=\left(x+5\right)\left(5y-14\right)$$

Use the distributive property to multiply y+3 by 5x-4y.

$$5yx-4y^{2}+15x-12y=\left(x+5\right)\left(5y-14\right)$$

Use the distributive property to multiply x+5 by 5y-14.

$$5yx-4y^{2}+15x-12y=5xy-14x+25y-70$$

Subtract 5xy from both sides.

$$5yx-4y^{2}+15x-12y-5xy=-14x+25y-70$$

Combine 5yx and -5xy to get 0.

$$-4y^{2}+15x-12y=-14x+25y-70$$

Add 14x to both sides.

$$-4y^{2}+15x-12y+14x=25y-70$$

Combine 15x and 14x to get 29x.

$$-4y^{2}+29x-12y=25y-70$$

Add 4y^{2} to both sides.

$$29x-12y=25y-70+4y^{2}$$

Add 12y to both sides.

$$29x=25y-70+4y^{2}+12y$$

Combine 25y and 12y to get 37y.

$$29x=37y-70+4y^{2}$$

The equation is in standard form.

$$29x=4y^{2}+37y-70$$

Divide both sides by 29.

$$\frac{29x}{29}=\frac{4y^{2}+37y-70}{29}$$

Dividing by 29 undoes the multiplication by 29.

$$x=\frac{4y^{2}+37y-70}{29}$$

Variable x cannot be equal to -5.

$$x=\frac{4y^{2}+37y-70}{29}\text{, }x\neq -5$$

Solve for y (complex solution)

$$y=\frac{-\sqrt{464x+2489}-37}{8}$$

$$y=\frac{\sqrt{464x+2489}-37}{8}\text{, }x\neq -5$$

Solve for y

$$y=\frac{-\sqrt{464x+2489}-37}{8}$$

$$y=\frac{\sqrt{464x+2489}-37}{8}\text{, }x\neq -5\text{ and }x\geq -\frac{2489}{464}$$