Solve 10/7=a+1/b+frac{1{c}}

$$\frac { 10 } { 7 } = a + \frac { 1 } { b + \frac { 1 } { c } }$$

Solve for a

$$a=\frac{10bc-7c+10}{7\left(bc+1\right)}$$

$$b\neq -\frac{1}{c}\text{ and }c\neq 0$$

Steps for Solving Linear Equation

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

$$\frac{10}{7}=a+\frac{1}{\frac{bc}{c}+\frac{1}{c}}$$

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

$$\frac{10}{7}=a+\frac{1}{\frac{bc+1}{c}}$$

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

$$\frac{10}{7}=a+\frac{c}{bc+1}$$

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

$$\frac{10}{7}=\frac{a\left(bc+1\right)}{bc+1}+\frac{c}{bc+1}$$

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

$$\frac{10}{7}=\frac{a\left(bc+1\right)+c}{bc+1}$$

Do the multiplications in a\left(bc+1\right)+c.

$$\frac{10}{7}=\frac{abc+a+c}{bc+1}$$

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

$$\frac{abc+a+c}{bc+1}=\frac{10}{7}$$

Multiply both sides of the equation by 7\left(bc+1\right), the least common multiple of bc+1,7.

$$7\left(abc+a+c\right)=10\left(bc+1\right)$$

Use the distributive property to multiply 7 by abc+a+c.

$$7abc+7a+7c=10\left(bc+1\right)$$

Use the distributive property to multiply 10 by bc+1.

$$7abc+7a+7c=10bc+10$$

Subtract 7c from both sides.

$$7abc+7a=10bc+10-7c$$

Combine all terms containing a.

$$\left(7bc+7\right)a=10bc+10-7c$$

The equation is in standard form.

$$\left(7bc+7\right)a=10bc-7c+10$$

Divide both sides by 7bc+7.

$$\frac{\left(7bc+7\right)a}{7bc+7}=\frac{10bc-7c+10}{7bc+7}$$

Dividing by 7bc+7 undoes the multiplication by 7bc+7.

$$a=\frac{10bc-7c+10}{7bc+7}$$

Divide 10bc-7c+10 by 7bc+7.

$$a=\frac{10bc-7c+10}{7\left(bc+1\right)}$$

Solve for b

$$b=-\frac{7a+7c-10}{c\left(7a-10\right)}$$

$$a\neq \frac{10}{7}\text{ and }c\neq 0$$