Solve 2/a+1/b=8/4+2b

$$\frac { 2 } { a } + \frac { 1 } { b } = \frac { 8 } { 4 + 2 b }$$

Solve for a

$$a=-\frac{2b\left(b+2\right)}{2-3b}$$

$$b\neq -2\text{ and }b\neq 0\text{ and }b\neq \frac{2}{3}$$

Steps for Solving Linear Equation

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2ab\left(b+2\right), the least common multiple of a,b,4+2b.

$$2b\left(b+2\right)\times 2+2a\left(b+2\right)=ab\times 8$$

Use the distributive property to multiply 2b by b+2.

$$\left(2b^{2}+4b\right)\times 2+2a\left(b+2\right)=ab\times 8$$

Use the distributive property to multiply 2b^{2}+4b by 2.

$$4b^{2}+8b+2a\left(b+2\right)=ab\times 8$$

Use the distributive property to multiply 2a by b+2.

$$4b^{2}+8b+2ab+4a=ab\times 8$$

Subtract ab\times 8 from both sides.

$$4b^{2}+8b+2ab+4a-ab\times 8=0$$

Combine 2ab and -ab\times 8 to get -6ab.

$$4b^{2}+8b-6ab+4a=0$$

Subtract 4b^{2} from both sides. Anything subtracted from zero gives its negation.

$$8b-6ab+4a=-4b^{2}$$

Subtract 8b from both sides.

$$-6ab+4a=-4b^{2}-8b$$

Combine all terms containing a.

$$\left(-6b+4\right)a=-4b^{2}-8b$$

The equation is in standard form.

$$\left(4-6b\right)a=-4b^{2}-8b$$

Divide both sides by -6b+4.

$$\frac{\left(4-6b\right)a}{4-6b}=-\frac{4b\left(b+2\right)}{4-6b}$$

Dividing by -6b+4 undoes the multiplication by -6b+4.

$$a=-\frac{4b\left(b+2\right)}{4-6b}$$

Divide -4b\left(2+b\right) by -6b+4.

$$a=-\frac{2b\left(b+2\right)}{2-3b}$$

Variable a cannot be equal to 0.

$$a=-\frac{2b\left(b+2\right)}{2-3b}\text{, }a\neq 0$$

Solve for b

$$b=\frac{\sqrt{\left(a-4\right)\left(9a-4\right)}+3a-4}{4}$$

$$b=\frac{-\sqrt{\left(a-4\right)\left(9a-4\right)}+3a-4}{4}\text{, }a\geq 4\text{ or }\left(a\neq 0\text{ and }a\leq \frac{4}{9}\right)$$