Solve +1/2+1{1+frac{1/2+frac{1{A}}}}=64/27

$$+ \frac { 1 } { 2 + \frac { 1 } { 1 + \frac { 1 } { 2 + \frac { 1 } { A } } } } = \frac { 64 } { 27 }$$

Solve for A

$$A=-\frac{165}{431}\approx -0.382830626$$

Steps for Solving Linear Equation

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{A}{A}.

$$\frac{1}{2+\frac{1}{1+\frac{1}{\frac{2A}{A}+\frac{1}{A}}}}=\frac{64}{27}$$

Since \frac{2A}{A} and \frac{1}{A} have the same denominator, add them by adding their numerators.

$$\frac{1}{2+\frac{1}{1+\frac{1}{\frac{2A+1}{A}}}}=\frac{64}{27}$$

Variable A cannot be equal to 0 since division by zero is not defined. Divide 1 by \frac{2A+1}{A} by multiplying 1 by the reciprocal of \frac{2A+1}{A}.

$$\frac{1}{2+\frac{1}{1+\frac{A}{2A+1}}}=\frac{64}{27}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2A+1}{2A+1}.

$$\frac{1}{2+\frac{1}{\frac{2A+1}{2A+1}+\frac{A}{2A+1}}}=\frac{64}{27}$$

Since \frac{2A+1}{2A+1} and \frac{A}{2A+1} have the same denominator, add them by adding their numerators.

$$\frac{1}{2+\frac{1}{\frac{2A+1+A}{2A+1}}}=\frac{64}{27}$$

Combine like terms in 2A+1+A.

$$\frac{1}{2+\frac{1}{\frac{3A+1}{2A+1}}}=\frac{64}{27}$$

Variable A cannot be equal to -\frac{1}{2} since division by zero is not defined. Divide 1 by \frac{3A+1}{2A+1} by multiplying 1 by the reciprocal of \frac{3A+1}{2A+1}.

$$\frac{1}{2+\frac{2A+1}{3A+1}}=\frac{64}{27}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{3A+1}{3A+1}.

$$\frac{1}{\frac{2\left(3A+1\right)}{3A+1}+\frac{2A+1}{3A+1}}=\frac{64}{27}$$

Since \frac{2\left(3A+1\right)}{3A+1} and \frac{2A+1}{3A+1} have the same denominator, add them by adding their numerators.

$$\frac{1}{\frac{2\left(3A+1\right)+2A+1}{3A+1}}=\frac{64}{27}$$

Do the multiplications in 2\left(3A+1\right)+2A+1.

$$\frac{1}{\frac{6A+2+2A+1}{3A+1}}=\frac{64}{27}$$

Combine like terms in 6A+2+2A+1.

$$\frac{1}{\frac{8A+3}{3A+1}}=\frac{64}{27}$$

Variable A cannot be equal to -\frac{1}{3} since division by zero is not defined. Divide 1 by \frac{8A+3}{3A+1} by multiplying 1 by the reciprocal of \frac{8A+3}{3A+1}.

$$\frac{3A+1}{8A+3}=\frac{64}{27}$$

Variable A cannot be equal to -\frac{3}{8} since division by zero is not defined. Multiply both sides of the equation by 27\left(8A+3\right), the least common multiple of 8A+3,27.

$$27\left(3A+1\right)=64\left(8A+3\right)$$

Use the distributive property to multiply 27 by 3A+1.

$$81A+27=64\left(8A+3\right)$$

Use the distributive property to multiply 64 by 8A+3.

$$81A+27=512A+192$$

Subtract 512A from both sides.

$$81A+27-512A=192$$

Combine 81A and -512A to get -431A.

$$-431A+27=192$$

Subtract 27 from both sides.

$$-431A=192-27$$

Subtract 27 from 192 to get 165.

$$-431A=165$$

Divide both sides by -431.

$$A=\frac{165}{-431}$$

Fraction \frac{165}{-431} can be rewritten as -\frac{165}{431} by extracting the negative sign.

$$A=-\frac{165}{431}$$