Solve 57/17=a+1/b+frac{c{d}}

$$\frac { 57 } { 17 } = a + \frac { 1 } { b + \frac { c } { d } }$$

Solve for a

$$a=\frac{57bd+57c-17d}{17\left(bd+c\right)}$$

$$c\neq -bd\text{ and }b\neq -\frac{c}{d}\text{ and }d\neq 0$$

Steps for Solving Linear Equation

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

$$\frac{57}{17}=a+\frac{1}{\frac{bd}{d}+\frac{c}{d}}$$

Since \frac{bd}{d} and \frac{c}{d} have the same denominator, add them by adding their numerators.

$$\frac{57}{17}=a+\frac{1}{\frac{bd+c}{d}}$$

Divide 1 by \frac{bd+c}{d} by multiplying 1 by the reciprocal of \frac{bd+c}{d}.

$$\frac{57}{17}=a+\frac{d}{bd+c}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{bd+c}{bd+c}.

$$\frac{57}{17}=\frac{a\left(bd+c\right)}{bd+c}+\frac{d}{bd+c}$$

Since \frac{a\left(bd+c\right)}{bd+c} and \frac{d}{bd+c} have the same denominator, add them by adding their numerators.

$$\frac{57}{17}=\frac{a\left(bd+c\right)+d}{bd+c}$$

Do the multiplications in a\left(bd+c\right)+d.

$$\frac{57}{17}=\frac{abd+ac+d}{bd+c}$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{abd+ac+d}{bd+c}=\frac{57}{17}$$

Multiply both sides of the equation by 17\left(bd+c\right), the least common multiple of bd+c,17.

$$17\left(abd+ac+d\right)=57\left(bd+c\right)$$

Use the distributive property to multiply 17 by abd+ac+d.

$$17abd+17ac+17d=57\left(bd+c\right)$$

Use the distributive property to multiply 57 by bd+c.

$$17abd+17ac+17d=57bd+57c$$

Subtract 17d from both sides.

$$17abd+17ac=57bd+57c-17d$$

Combine all terms containing a.

$$\left(17bd+17c\right)a=57bd+57c-17d$$

Divide both sides by 17bd+17c.

$$\frac{\left(17bd+17c\right)a}{17bd+17c}=\frac{57bd+57c-17d}{17bd+17c}$$

Dividing by 17bd+17c undoes the multiplication by 17bd+17c.

$$a=\frac{57bd+57c-17d}{17bd+17c}$$

Divide 57bd+57c-17d by 17bd+17c.

$$a=\frac{57bd+57c-17d}{17\left(bd+c\right)}$$

Solve for b (complex solution)

$$b=-\frac{17ac-57c+17d}{d\left(17a-57\right)}$$

$$d\neq 0\text{ and }a\neq \frac{57}{17}$$

Solve for b

$$b=-\frac{17ac-57c+17d}{d\left(17a-57\right)}$$

$$a\neq \frac{57}{17}\text{ and }d\neq 0$$