Solve a/6-1/6-b=a/10-1/10-b

$$\frac { a } { 6 } - \frac { 1 } { 6 - b } = \frac { a } { 10 } - \frac { 1 } { 10 - b }$$

Solve for a

$$a=\frac{60}{\left(b-10\right)\left(b-6\right)}$$

$$b\neq 6\text{ and }b\neq 10$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 30\left(b-10\right)\left(b-6\right), the least common multiple of 6,6-b,10,10-b.

$$5\left(b-10\right)\left(b-6\right)a-\left(300-30b\right)=3\left(b-10\right)\left(b-6\right)a-\left(180-30b\right)$$

Use the distributive property to multiply 5 by b-10.

$$\left(5b-50\right)\left(b-6\right)a-\left(300-30b\right)=3\left(b-10\right)\left(b-6\right)a-\left(180-30b\right)$$

Use the distributive property to multiply 5b-50 by b-6 and combine like terms.

$$\left(5b^{2}-80b+300\right)a-\left(300-30b\right)=3\left(b-10\right)\left(b-6\right)a-\left(180-30b\right)$$

Use the distributive property to multiply 5b^{2}-80b+300 by a.

$$5b^{2}a-80ba+300a-\left(300-30b\right)=3\left(b-10\right)\left(b-6\right)a-\left(180-30b\right)$$

To find the opposite of 300-30b, find the opposite of each term.

$$5b^{2}a-80ba+300a-300+30b=3\left(b-10\right)\left(b-6\right)a-\left(180-30b\right)$$

Use the distributive property to multiply 3 by b-10.

$$5b^{2}a-80ba+300a-300+30b=\left(3b-30\right)\left(b-6\right)a-\left(180-30b\right)$$

Use the distributive property to multiply 3b-30 by b-6 and combine like terms.

$$5b^{2}a-80ba+300a-300+30b=\left(3b^{2}-48b+180\right)a-\left(180-30b\right)$$

Use the distributive property to multiply 3b^{2}-48b+180 by a.

$$5b^{2}a-80ba+300a-300+30b=3b^{2}a-48ba+180a-\left(180-30b\right)$$

To find the opposite of 180-30b, find the opposite of each term.

$$5b^{2}a-80ba+300a-300+30b=3b^{2}a-48ba+180a-180+30b$$

Subtract 3b^{2}a from both sides.

$$5b^{2}a-80ba+300a-300+30b-3b^{2}a=-48ba+180a-180+30b$$

Combine 5b^{2}a and -3b^{2}a to get 2b^{2}a.

$$2b^{2}a-80ba+300a-300+30b=-48ba+180a-180+30b$$

Add 48ba to both sides.

$$2b^{2}a-80ba+300a-300+30b+48ba=180a-180+30b$$

Combine -80ba and 48ba to get -32ba.

$$2b^{2}a-32ba+300a-300+30b=180a-180+30b$$

Subtract 180a from both sides.

$$2b^{2}a-32ba+300a-300+30b-180a=-180+30b$$

Combine 300a and -180a to get 120a.

$$2b^{2}a-32ba+120a-300+30b=-180+30b$$

Add 300 to both sides.

$$2b^{2}a-32ba+120a+30b=-180+30b+300$$

Add -180 and 300 to get 120.

$$2b^{2}a-32ba+120a+30b=120+30b$$

Subtract 30b from both sides.

$$2b^{2}a-32ba+120a=120+30b-30b$$

Combine 30b and -30b to get 0.

$$2b^{2}a-32ba+120a=120$$

Combine all terms containing a.

$$\left(2b^{2}-32b+120\right)a=120$$

Divide both sides by 2b^{2}-32b+120.

$$\frac{\left(2b^{2}-32b+120\right)a}{2b^{2}-32b+120}=\frac{120}{2b^{2}-32b+120}$$

Dividing by 2b^{2}-32b+120 undoes the multiplication by 2b^{2}-32b+120.

$$a=\frac{120}{2b^{2}-32b+120}$$

Divide 120 by 2b^{2}-32b+120.

$$a=\frac{60}{\left(b-10\right)\left(b-6\right)}$$

Solve for b (complex solution)

$$b=-\frac{2\sqrt{a^{2}+15a}}{a}+8$$

$$b=\frac{2\sqrt{a^{2}+15a}}{a}+8\text{, }a\neq 0$$

Solve for b

$$\left\{\begin{matrix}b=-2\sqrt{1+\frac{15}{a}}+8\text{; }b=2\sqrt{1+\frac{15}{a}}+8\text{, }&a>0\\b=-\frac{2\sqrt{a^{2}+15a}}{a}+8\text{; }b=\frac{2\sqrt{a^{2}+15a}}{a}+8\text{, }&a\leq -15\end{matrix}\right.$$