Solve 2b-5/2x-6=5x-4/2x-10+1/2

$$\frac { 2 b - 5 } { 2 x - 6 } = \frac { 5 x - 4 } { 2 x - 10 } + \frac { 1 } { 2 }$$

Solve for b

$$b=-\frac{-3x^{2}+11x-1}{x-5}$$

$$x\neq 3\text{ and }x\neq 5$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2\left(x-5\right)\left(x-3\right), the least common multiple of 2x-6,2x-10,2.

$$\left(x-5\right)\left(2b-5\right)=\left(x-3\right)\left(5x-4\right)+2\left(x-5\right)\left(x-3\right)\times \frac{1}{2}$$

Use the distributive property to multiply x-5 by 2b-5.

$$2xb-5x-10b+25=\left(x-3\right)\left(5x-4\right)+2\left(x-5\right)\left(x-3\right)\times \frac{1}{2}$$

Use the distributive property to multiply x-3 by 5x-4 and combine like terms.

$$2xb-5x-10b+25=5x^{2}-19x+12+2\left(x-5\right)\left(x-3\right)\times \frac{1}{2}$$

Multiply 2 and \frac{1}{2} to get 1.

$$2xb-5x-10b+25=5x^{2}-19x+12+\left(x-5\right)\left(x-3\right)$$

Use the distributive property to multiply x-5 by x-3 and combine like terms.

$$2xb-5x-10b+25=5x^{2}-19x+12+x^{2}-8x+15$$

Combine 5x^{2} and x^{2} to get 6x^{2}.

$$2xb-5x-10b+25=6x^{2}-19x+12-8x+15$$

Combine -19x and -8x to get -27x.

$$2xb-5x-10b+25=6x^{2}-27x+12+15$$

Add 12 and 15 to get 27.

$$2xb-5x-10b+25=6x^{2}-27x+27$$

Add 5x to both sides.

$$2xb-10b+25=6x^{2}-27x+27+5x$$

Combine -27x and 5x to get -22x.

$$2xb-10b+25=6x^{2}-22x+27$$

Subtract 25 from both sides.

$$2xb-10b=6x^{2}-22x+27-25$$

Subtract 25 from 27 to get 2.

$$2xb-10b=6x^{2}-22x+2$$

Combine all terms containing b.

$$\left(2x-10\right)b=6x^{2}-22x+2$$

Divide both sides by 2x-10.

$$\frac{\left(2x-10\right)b}{2x-10}=\frac{6x^{2}-22x+2}{2x-10}$$

Dividing by 2x-10 undoes the multiplication by 2x-10.

$$b=\frac{6x^{2}-22x+2}{2x-10}$$

Divide 6x^{2}-22x+2 by 2x-10.

$$b=\frac{3x^{2}-11x+1}{x-5}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{-\sqrt{b^{2}-38b+109}+b+11}{6}\text{, }&\text{unconditionally}\\x=\frac{\sqrt{b^{2}-38b+109}+b+11}{6}\text{, }&b\neq \frac{5}{2}\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{b^{2}-38b+109}+b+11}{6}\text{, }&b\geq 6\sqrt{7}+19\text{ or }\left(b\neq \frac{5}{2}\text{ and }b\leq 19-6\sqrt{7}\right)\\x=\frac{-\sqrt{b^{2}-38b+109}+b+11}{6}\text{, }&b\geq 6\sqrt{7}+19\text{ or }b\leq 19-6\sqrt{7}\end{matrix}\right.$$