Solve 1/x-10+1/x-6=1/x-7+1/x-9

$$\frac { 1 } { x - 10 } + \frac { 1 } { x - 6 } = \frac { 1 } { x - 7 } + \frac { 1 } { x - 9 }$$

Solve for x

$$x=8$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values 6,7,9,10 since division by zero is not defined. Multiply both sides of the equation by \left(x-10\right)\left(x-9\right)\left(x-7\right)\left(x-6\right), the least common multiple of x-10,x-6,x-7,x-9.

$$\left(x-9\right)\left(x-7\right)\left(x-6\right)+\left(x-10\right)\left(x-9\right)\left(x-7\right)=\left(x-10\right)\left(x-9\right)\left(x-6\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)$$

Use the distributive property to multiply x-9 by x-7 and combine like terms.

$$\left(x^{2}-16x+63\right)\left(x-6\right)+\left(x-10\right)\left(x-9\right)\left(x-7\right)=\left(x-10\right)\left(x-9\right)\left(x-6\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)$$

Use the distributive property to multiply x^{2}-16x+63 by x-6 and combine like terms.

$$x^{3}-22x^{2}+159x-378+\left(x-10\right)\left(x-9\right)\left(x-7\right)=\left(x-10\right)\left(x-9\right)\left(x-6\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)$$

Use the distributive property to multiply x-10 by x-9 and combine like terms.

$$x^{3}-22x^{2}+159x-378+\left(x^{2}-19x+90\right)\left(x-7\right)=\left(x-10\right)\left(x-9\right)\left(x-6\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)$$

Use the distributive property to multiply x^{2}-19x+90 by x-7 and combine like terms.

$$x^{3}-22x^{2}+159x-378+x^{3}-26x^{2}+223x-630=\left(x-10\right)\left(x-9\right)\left(x-6\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)$$

Combine x^{3} and x^{3} to get 2x^{3}.

$$2x^{3}-22x^{2}+159x-378-26x^{2}+223x-630=\left(x-10\right)\left(x-9\right)\left(x-6\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)$$

Combine -22x^{2} and -26x^{2} to get -48x^{2}.

$$2x^{3}-48x^{2}+159x-378+223x-630=\left(x-10\right)\left(x-9\right)\left(x-6\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)$$

Combine 159x and 223x to get 382x.

$$2x^{3}-48x^{2}+382x-378-630=\left(x-10\right)\left(x-9\right)\left(x-6\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)$$

Subtract 630 from -378 to get -1008.

$$2x^{3}-48x^{2}+382x-1008=\left(x-10\right)\left(x-9\right)\left(x-6\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)$$

Use the distributive property to multiply x-10 by x-9 and combine like terms.

$$2x^{3}-48x^{2}+382x-1008=\left(x^{2}-19x+90\right)\left(x-6\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)$$

Use the distributive property to multiply x^{2}-19x+90 by x-6 and combine like terms.

$$2x^{3}-48x^{2}+382x-1008=x^{3}-25x^{2}+204x-540+\left(x-10\right)\left(x-7\right)\left(x-6\right)$$

Use the distributive property to multiply x-10 by x-7 and combine like terms.

$$2x^{3}-48x^{2}+382x-1008=x^{3}-25x^{2}+204x-540+\left(x^{2}-17x+70\right)\left(x-6\right)$$

Use the distributive property to multiply x^{2}-17x+70 by x-6 and combine like terms.

$$2x^{3}-48x^{2}+382x-1008=x^{3}-25x^{2}+204x-540+x^{3}-23x^{2}+172x-420$$

Combine x^{3} and x^{3} to get 2x^{3}.

$$2x^{3}-48x^{2}+382x-1008=2x^{3}-25x^{2}+204x-540-23x^{2}+172x-420$$

Combine -25x^{2} and -23x^{2} to get -48x^{2}.

$$2x^{3}-48x^{2}+382x-1008=2x^{3}-48x^{2}+204x-540+172x-420$$

Combine 204x and 172x to get 376x.

$$2x^{3}-48x^{2}+382x-1008=2x^{3}-48x^{2}+376x-540-420$$

Subtract 420 from -540 to get -960.

$$2x^{3}-48x^{2}+382x-1008=2x^{3}-48x^{2}+376x-960$$

Subtract 2x^{3} from both sides.

$$2x^{3}-48x^{2}+382x-1008-2x^{3}=-48x^{2}+376x-960$$

Combine 2x^{3} and -2x^{3} to get 0.

$$-48x^{2}+382x-1008=-48x^{2}+376x-960$$

Add 48x^{2} to both sides.

$$-48x^{2}+382x-1008+48x^{2}=376x-960$$

Combine -48x^{2} and 48x^{2} to get 0.

$$382x-1008=376x-960$$

Subtract 376x from both sides.

$$382x-1008-376x=-960$$

Combine 382x and -376x to get 6x.

$$6x-1008=-960$$

Add 1008 to both sides.

$$6x=-960+1008$$

Add -960 and 1008 to get 48.

$$6x=48$$

Divide both sides by 6.

$$x=\frac{48}{6}$$

Divide 48 by 6 to get 8.

$$x=8$$