Solve x-8/x-10+x-4/x-6=x-5/x-7+x-7/x-9

$$\frac { x - 8 } { x - 10 } + \frac { x - 4 } { x - 6 } = \frac { x - 5 } { x - 7 } + \frac { x - 7 } { x - 9 }$$

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.

$$\left(x-9\right)\left(x-7\right)\left(x-6\right)\left(x-8\right)+\left(x-10\right)\left(x-9\right)\left(x-7\right)\left(x-4\right)=\left(x-10\right)\left(x-9\right)\left(x-6\right)\left(x-5\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)\left(x-7\right)$$

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

$$\left(x^{2}-16x+63\right)\left(x-6\right)\left(x-8\right)+\left(x-10\right)\left(x-9\right)\left(x-7\right)\left(x-4\right)=\left(x-10\right)\left(x-9\right)\left(x-6\right)\left(x-5\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)\left(x-7\right)$$

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

$$\left(x^{3}-22x^{2}+159x-378\right)\left(x-8\right)+\left(x-10\right)\left(x-9\right)\left(x-7\right)\left(x-4\right)=\left(x-10\right)\left(x-9\right)\left(x-6\right)\left(x-5\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)\left(x-7\right)$$

Use the distributive property to multiply x^{3}-22x^{2}+159x-378 by x-8 and combine like terms.

$$x^{4}-30x^{3}+335x^{2}-1650x+3024+\left(x-10\right)\left(x-9\right)\left(x-7\right)\left(x-4\right)=\left(x-10\right)\left(x-9\right)\left(x-6\right)\left(x-5\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)\left(x-7\right)$$

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

$$x^{4}-30x^{3}+335x^{2}-1650x+3024+\left(x^{2}-19x+90\right)\left(x-7\right)\left(x-4\right)=\left(x-10\right)\left(x-9\right)\left(x-6\right)\left(x-5\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)\left(x-7\right)$$

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

$$x^{4}-30x^{3}+335x^{2}-1650x+3024+\left(x^{3}-26x^{2}+223x-630\right)\left(x-4\right)=\left(x-10\right)\left(x-9\right)\left(x-6\right)\left(x-5\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)\left(x-7\right)$$

Use the distributive property to multiply x^{3}-26x^{2}+223x-630 by x-4 and combine like terms.

$$x^{4}-30x^{3}+335x^{2}-1650x+3024+x^{4}-30x^{3}+327x^{2}-1522x+2520=\left(x-10\right)\left(x-9\right)\left(x-6\right)\left(x-5\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)\left(x-7\right)$$

Combine x^{4} and x^{4} to get 2x^{4}.

$$2x^{4}-30x^{3}+335x^{2}-1650x+3024-30x^{3}+327x^{2}-1522x+2520=\left(x-10\right)\left(x-9\right)\left(x-6\right)\left(x-5\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)\left(x-7\right)$$

Combine -30x^{3} and -30x^{3} to get -60x^{3}.

$$2x^{4}-60x^{3}+335x^{2}-1650x+3024+327x^{2}-1522x+2520=\left(x-10\right)\left(x-9\right)\left(x-6\right)\left(x-5\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)\left(x-7\right)$$

Combine 335x^{2} and 327x^{2} to get 662x^{2}.

$$2x^{4}-60x^{3}+662x^{2}-1650x+3024-1522x+2520=\left(x-10\right)\left(x-9\right)\left(x-6\right)\left(x-5\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)\left(x-7\right)$$

Combine -1650x and -1522x to get -3172x.

$$2x^{4}-60x^{3}+662x^{2}-3172x+3024+2520=\left(x-10\right)\left(x-9\right)\left(x-6\right)\left(x-5\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)\left(x-7\right)$$

Add 3024 and 2520 to get 5544.

$$2x^{4}-60x^{3}+662x^{2}-3172x+5544=\left(x-10\right)\left(x-9\right)\left(x-6\right)\left(x-5\right)+\left(x-10\right)\left(x-7\right)\left(x-6\right)\left(x-7\right)$$

Multiply x-7 and x-7 to get \left(x-7\right)^{2}.

$$2x^{4}-60x^{3}+662x^{2}-3172x+5544=\left(x-10\right)\left(x-9\right)\left(x-6\right)\left(x-5\right)+\left(x-10\right)\left(x-7\right)^{2}\left(x-6\right)$$

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

$$2x^{4}-60x^{3}+662x^{2}-3172x+5544=\left(x^{2}-19x+90\right)\left(x-6\right)\left(x-5\right)+\left(x-10\right)\left(x-7\right)^{2}\left(x-6\right)$$

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

$$2x^{4}-60x^{3}+662x^{2}-3172x+5544=\left(x^{3}-25x^{2}+204x-540\right)\left(x-5\right)+\left(x-10\right)\left(x-7\right)^{2}\left(x-6\right)$$

Use the distributive property to multiply x^{3}-25x^{2}+204x-540 by x-5 and combine like terms.

$$2x^{4}-60x^{3}+662x^{2}-3172x+5544=x^{4}-30x^{3}+329x^{2}-1560x+2700+\left(x-10\right)\left(x-7\right)^{2}\left(x-6\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-7\right)^{2}.

$$2x^{4}-60x^{3}+662x^{2}-3172x+5544=x^{4}-30x^{3}+329x^{2}-1560x+2700+\left(x-10\right)\left(x^{2}-14x+49\right)\left(x-6\right)$$

Use the distributive property to multiply x-10 by x^{2}-14x+49 and combine like terms.

$$2x^{4}-60x^{3}+662x^{2}-3172x+5544=x^{4}-30x^{3}+329x^{2}-1560x+2700+\left(x^{3}-24x^{2}+189x-490\right)\left(x-6\right)$$

Use the distributive property to multiply x^{3}-24x^{2}+189x-490 by x-6 and combine like terms.

$$2x^{4}-60x^{3}+662x^{2}-3172x+5544=x^{4}-30x^{3}+329x^{2}-1560x+2700+x^{4}-30x^{3}+333x^{2}-1624x+2940$$

Combine x^{4} and x^{4} to get 2x^{4}.

$$2x^{4}-60x^{3}+662x^{2}-3172x+5544=2x^{4}-30x^{3}+329x^{2}-1560x+2700-30x^{3}+333x^{2}-1624x+2940$$

Combine -30x^{3} and -30x^{3} to get -60x^{3}.

$$2x^{4}-60x^{3}+662x^{2}-3172x+5544=2x^{4}-60x^{3}+329x^{2}-1560x+2700+333x^{2}-1624x+2940$$

Combine 329x^{2} and 333x^{2} to get 662x^{2}.

$$2x^{4}-60x^{3}+662x^{2}-3172x+5544=2x^{4}-60x^{3}+662x^{2}-1560x+2700-1624x+2940$$

Combine -1560x and -1624x to get -3184x.

$$2x^{4}-60x^{3}+662x^{2}-3172x+5544=2x^{4}-60x^{3}+662x^{2}-3184x+2700+2940$$

Add 2700 and 2940 to get 5640.

$$2x^{4}-60x^{3}+662x^{2}-3172x+5544=2x^{4}-60x^{3}+662x^{2}-3184x+5640$$

Subtract 2x^{4} from both sides.

$$2x^{4}-60x^{3}+662x^{2}-3172x+5544-2x^{4}=-60x^{3}+662x^{2}-3184x+5640$$

Combine 2x^{4} and -2x^{4} to get 0.

$$-60x^{3}+662x^{2}-3172x+5544=-60x^{3}+662x^{2}-3184x+5640$$

Add 60x^{3} to both sides.

$$-60x^{3}+662x^{2}-3172x+5544+60x^{3}=662x^{2}-3184x+5640$$

Combine -60x^{3} and 60x^{3} to get 0.

$$662x^{2}-3172x+5544=662x^{2}-3184x+5640$$

Subtract 662x^{2} from both sides.

$$662x^{2}-3172x+5544-662x^{2}=-3184x+5640$$

Combine 662x^{2} and -662x^{2} to get 0.

$$-3172x+5544=-3184x+5640$$

Add 3184x to both sides.

$$-3172x+5544+3184x=5640$$

Combine -3172x and 3184x to get 12x.

$$12x+5544=5640$$

Subtract 5544 from both sides.

$$12x=5640-5544$$

Subtract 5544 from 5640 to get 96.

$$12x=96$$

Divide both sides by 12.

$$x=\frac{96}{12}$$

Divide 96 by 12 to get 8.

$$x=8$$