Solve 30x*(y+20-x)/(3x+5)y(20-x)=6*(x+5)/3x+5

$$\frac{ 30x \times (y+20-x) }{ (3x+5)y(20-x) } = \frac{ 6 \times (x+5) }{ 3x+5 }$$

Solve for y

$$y=-\frac{5x\left(20-x\right)}{x^{2}-10x-100}$$

$$x\neq 20\text{ and }x\neq 0\text{ and }x\neq 5\sqrt{5}+5\text{ and }x\neq 5-5\sqrt{5}\text{ and }x\neq -\frac{5}{3}$$

Steps for Solving Linear Equation

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by y\left(x-20\right)\left(3x+5\right), the least common multiple of \left(3x+5\right)y\left(20-x\right),3x+5.

$$-30x\left(y+20-x\right)=y\left(x-20\right)\times 6\left(x+5\right)$$

Use the distributive property to multiply -30x by y+20-x.

$$-30xy-600x+30x^{2}=y\left(x-20\right)\times 6\left(x+5\right)$$

Use the distributive property to multiply y by x-20.

$$-30xy-600x+30x^{2}=\left(yx-20y\right)\times 6\left(x+5\right)$$

Use the distributive property to multiply yx-20y by 6.

$$-30xy-600x+30x^{2}=\left(6yx-120y\right)\left(x+5\right)$$

Use the distributive property to multiply 6yx-120y by x+5 and combine like terms.

$$-30xy-600x+30x^{2}=6yx^{2}-90yx-600y$$

Subtract 6yx^{2} from both sides.

$$-30xy-600x+30x^{2}-6yx^{2}=-90yx-600y$$

Add 90yx to both sides.

$$-30xy-600x+30x^{2}-6yx^{2}+90yx=-600y$$

Add 600y to both sides.

$$-30xy-600x+30x^{2}-6yx^{2}+90yx+600y=0$$

Combine -30xy and 90yx to get 60xy.

$$60xy-600x+30x^{2}-6yx^{2}+600y=0$$

Add 600x to both sides. Anything plus zero gives itself.

$$60xy+30x^{2}-6yx^{2}+600y=600x$$

Subtract 30x^{2} from both sides.

$$60xy-6yx^{2}+600y=600x-30x^{2}$$

Combine all terms containing y.

$$\left(60x-6x^{2}+600\right)y=600x-30x^{2}$$

The equation is in standard form.

$$\left(600+60x-6x^{2}\right)y=600x-30x^{2}$$

Divide both sides by 60x-6x^{2}+600.

$$\frac{\left(600+60x-6x^{2}\right)y}{600+60x-6x^{2}}=\frac{30x\left(20-x\right)}{600+60x-6x^{2}}$$

Dividing by 60x-6x^{2}+600 undoes the multiplication by 60x-6x^{2}+600.

$$y=\frac{30x\left(20-x\right)}{600+60x-6x^{2}}$$

Divide 30x\left(20-x\right) by 60x-6x^{2}+600.

$$y=\frac{5x\left(20-x\right)}{100+10x-x^{2}}$$

Variable y cannot be equal to 0.

$$y=\frac{5x\left(20-x\right)}{100+10x-x^{2}}\text{, }y\neq 0$$

Solve for x

$$\left\{\begin{matrix}x=\frac{5\left(-\sqrt{5\left(y^{2}-8y+20\right)}+y-10\right)}{y-5}\text{, }&y\neq 0\text{ and }y\neq 5\\x=\frac{5\left(\sqrt{5\left(y^{2}-8y+20\right)}+y-10\right)}{y-5}\text{, }&y\neq -\frac{65}{29}\text{ and }y\neq 5\text{ and }y\neq 0\\x=10\text{, }&y=5\end{matrix}\right.$$