Solve 3b/2y+3-b-y/x-5=1

$$\frac { 3 b } { 2 y + 3 } - \frac { b - y } { x - 5 } = 1$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=\frac{\left(x-y-5\right)\left(2y+3\right)}{3x-2y-18}\text{, }&y\neq \frac{3x}{2}-9\text{ and }y\neq -\frac{3}{2}\text{ and }x\neq 5\\b\in \mathrm{C}\text{, }&y=3\text{ and }x=8\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-5\right)\left(2y+3\right), the least common multiple of 2y+3,x-5.

$$\left(x-5\right)\times 3b-\left(2y+3\right)\left(b-y\right)=\left(x-5\right)\left(2y+3\right)$$

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

$$\left(3x-15\right)b-\left(2y+3\right)\left(b-y\right)=\left(x-5\right)\left(2y+3\right)$$

Use the distributive property to multiply 3x-15 by b.

$$3xb-15b-\left(2y+3\right)\left(b-y\right)=\left(x-5\right)\left(2y+3\right)$$

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

$$3xb-15b-\left(2yb-2y^{2}+3b-3y\right)=\left(x-5\right)\left(2y+3\right)$$

To find the opposite of 2yb-2y^{2}+3b-3y, find the opposite of each term.

$$3xb-15b-2yb+2y^{2}-3b+3y=\left(x-5\right)\left(2y+3\right)$$

Combine -15b and -3b to get -18b.

$$3xb-18b-2yb+2y^{2}+3y=\left(x-5\right)\left(2y+3\right)$$

Use the distributive property to multiply x-5 by 2y+3.

$$3xb-18b-2yb+2y^{2}+3y=2xy+3x-10y-15$$

Subtract 2y^{2} from both sides.

$$3xb-18b-2yb+3y=2xy+3x-10y-15-2y^{2}$$

Subtract 3y from both sides.

$$3xb-18b-2yb=2xy+3x-10y-15-2y^{2}-3y$$

Combine -10y and -3y to get -13y.

$$3xb-18b-2yb=2xy+3x-13y-15-2y^{2}$$

Combine all terms containing b.

$$\left(3x-18-2y\right)b=2xy+3x-13y-15-2y^{2}$$

The equation is in standard form.

$$\left(3x-2y-18\right)b=2xy+3x-2y^{2}-13y-15$$

Divide both sides by 3x-2y-18.

$$\frac{\left(3x-2y-18\right)b}{3x-2y-18}=\frac{\left(x-y-5\right)\left(2y+3\right)}{3x-2y-18}$$

Dividing by 3x-2y-18 undoes the multiplication by 3x-2y-18.

$$b=\frac{\left(x-y-5\right)\left(2y+3\right)}{3x-2y-18}$$

Solve for b

$$\left\{\begin{matrix}b=\frac{\left(x-y-5\right)\left(2y+3\right)}{3x-2y-18}\text{, }&y\neq \frac{3x}{2}-9\text{ and }y\neq -\frac{3}{2}\text{ and }x\neq 5\\b\in \mathrm{R}\text{, }&y=3\text{ and }x=8\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-5\right)\left(2y+3\right), the least common multiple of 2y+3,x-5.

$$\left(x-5\right)\times 3b-\left(2y+3\right)\left(b-y\right)=\left(x-5\right)\left(2y+3\right)$$

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

$$\left(3x-15\right)b-\left(2y+3\right)\left(b-y\right)=\left(x-5\right)\left(2y+3\right)$$

Use the distributive property to multiply 3x-15 by b.

$$3xb-15b-\left(2y+3\right)\left(b-y\right)=\left(x-5\right)\left(2y+3\right)$$

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

$$3xb-15b-\left(2yb-2y^{2}+3b-3y\right)=\left(x-5\right)\left(2y+3\right)$$

To find the opposite of 2yb-2y^{2}+3b-3y, find the opposite of each term.

$$3xb-15b-2yb+2y^{2}-3b+3y=\left(x-5\right)\left(2y+3\right)$$

Combine -15b and -3b to get -18b.

$$3xb-18b-2yb+2y^{2}+3y=\left(x-5\right)\left(2y+3\right)$$

Use the distributive property to multiply x-5 by 2y+3.

$$3xb-18b-2yb+2y^{2}+3y=2xy+3x-10y-15$$

Subtract 2y^{2} from both sides.

$$3xb-18b-2yb+3y=2xy+3x-10y-15-2y^{2}$$

Subtract 3y from both sides.

$$3xb-18b-2yb=2xy+3x-10y-15-2y^{2}-3y$$

Combine -10y and -3y to get -13y.

$$3xb-18b-2yb=2xy+3x-13y-15-2y^{2}$$

Combine all terms containing b.

$$\left(3x-18-2y\right)b=2xy+3x-13y-15-2y^{2}$$

The equation is in standard form.

$$\left(3x-2y-18\right)b=2xy+3x-2y^{2}-13y-15$$

Divide both sides by 3x-2y-18.

$$\frac{\left(3x-2y-18\right)b}{3x-2y-18}=\frac{\left(x-y-5\right)\left(2y+3\right)}{3x-2y-18}$$

Dividing by 3x-2y-18 undoes the multiplication by 3x-2y-18.

$$b=\frac{\left(x-y-5\right)\left(2y+3\right)}{3x-2y-18}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{2y^{2}-2by+13y-18b+15}{-2y+3b-3}\text{, }&y\neq -\frac{3}{2}\text{ and }b\neq y\text{ and }b\neq \frac{2y}{3}+1\\x\neq 5\text{, }&y=3\text{ and }b=3\end{matrix}\right.$$

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(2y+3\right), the least common multiple of 2y+3,x-5.

$$\left(x-5\right)\times 3b-\left(2y+3\right)\left(b-y\right)=\left(x-5\right)\left(2y+3\right)$$

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

$$\left(3x-15\right)b-\left(2y+3\right)\left(b-y\right)=\left(x-5\right)\left(2y+3\right)$$

Use the distributive property to multiply 3x-15 by b.

$$3xb-15b-\left(2y+3\right)\left(b-y\right)=\left(x-5\right)\left(2y+3\right)$$

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

$$3xb-15b-\left(2yb-2y^{2}+3b-3y\right)=\left(x-5\right)\left(2y+3\right)$$

To find the opposite of 2yb-2y^{2}+3b-3y, find the opposite of each term.

$$3xb-15b-2yb+2y^{2}-3b+3y=\left(x-5\right)\left(2y+3\right)$$

Combine -15b and -3b to get -18b.

$$3xb-18b-2yb+2y^{2}+3y=\left(x-5\right)\left(2y+3\right)$$

Use the distributive property to multiply x-5 by 2y+3.

$$3xb-18b-2yb+2y^{2}+3y=2xy+3x-10y-15$$

Subtract 2xy from both sides.

$$3xb-18b-2yb+2y^{2}+3y-2xy=3x-10y-15$$

Subtract 3x from both sides.

$$3xb-18b-2yb+2y^{2}+3y-2xy-3x=-10y-15$$

Add 18b to both sides.

$$3xb-2yb+2y^{2}+3y-2xy-3x=-10y-15+18b$$

Add 2yb to both sides.

$$3xb+2y^{2}+3y-2xy-3x=-10y-15+18b+2yb$$

Subtract 2y^{2} from both sides.

$$3xb+3y-2xy-3x=-10y-15+18b+2yb-2y^{2}$$

Subtract 3y from both sides.

$$3xb-2xy-3x=-10y-15+18b+2yb-2y^{2}-3y$$

Combine -10y and -3y to get -13y.

$$3xb-2xy-3x=-13y-15+18b+2yb-2y^{2}$$

Combine all terms containing x.

$$\left(3b-2y-3\right)x=-13y-15+18b+2yb-2y^{2}$$

The equation is in standard form.

$$\left(-2y+3b-3\right)x=-2y^{2}+2by-13y+18b-15$$

Divide both sides by -2y+3b-3.

$$\frac{\left(-2y+3b-3\right)x}{-2y+3b-3}=\frac{-2y^{2}+2by-13y+18b-15}{-2y+3b-3}$$

Dividing by -2y+3b-3 undoes the multiplication by -2y+3b-3.

$$x=\frac{-2y^{2}+2by-13y+18b-15}{-2y+3b-3}$$

Variable x cannot be equal to 5.

$$x=\frac{-2y^{2}+2by-13y+18b-15}{-2y+3b-3}\text{, }x\neq 5$$