Solve x-1/x-2-x-2/x-3=x-5/x-6-x-6/x-7

$$\frac{ x-1 }{ x-2 } - \frac{ x-2 }{ x-3 } = \frac{ x-5 }{ x-6 } - \frac{ x-6 }{ x-7 }$$

Solve for x

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values 2,3,6,7 since division by zero is not defined. Multiply both sides of the equation by \left(x-7\right)\left(x-6\right)\left(x-3\right)\left(x-2\right), the least common multiple of x-2,x-3,x-6,x-7.

$$\left(x-7\right)\left(x-6\right)\left(x-3\right)\left(x-1\right)-\left(x-7\right)\left(x-6\right)\left(x-2\right)\left(x-2\right)=\left(x-7\right)\left(x-3\right)\left(x-2\right)\left(x-5\right)-\left(x-6\right)\left(x-3\right)\left(x-2\right)\left(x-6\right)$$

Use the distributive property to multiply x-7 by x-6 and combine like terms.

$$\left(x^{2}-13x+42\right)\left(x-3\right)\left(x-1\right)-\left(x-7\right)\left(x-6\right)\left(x-2\right)\left(x-2\right)=\left(x-7\right)\left(x-3\right)\left(x-2\right)\left(x-5\right)-\left(x-6\right)\left(x-3\right)\left(x-2\right)\left(x-6\right)$$

Use the distributive property to multiply x^{2}-13x+42 by x-3 and combine like terms.

$$\left(x^{3}-16x^{2}+81x-126\right)\left(x-1\right)-\left(x-7\right)\left(x-6\right)\left(x-2\right)\left(x-2\right)=\left(x-7\right)\left(x-3\right)\left(x-2\right)\left(x-5\right)-\left(x-6\right)\left(x-3\right)\left(x-2\right)\left(x-6\right)$$

Use the distributive property to multiply x^{3}-16x^{2}+81x-126 by x-1 and combine like terms.

$$x^{4}-17x^{3}+97x^{2}-207x+126-\left(x-7\right)\left(x-6\right)\left(x-2\right)\left(x-2\right)=\left(x-7\right)\left(x-3\right)\left(x-2\right)\left(x-5\right)-\left(x-6\right)\left(x-3\right)\left(x-2\right)\left(x-6\right)$$

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

$$x^{4}-17x^{3}+97x^{2}-207x+126-\left(x-7\right)\left(x-6\right)\left(x-2\right)^{2}=\left(x-7\right)\left(x-3\right)\left(x-2\right)\left(x-5\right)-\left(x-6\right)\left(x-3\right)\left(x-2\right)\left(x-6\right)$$

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

$$x^{4}-17x^{3}+97x^{2}-207x+126-\left(x-7\right)\left(x-6\right)\left(x^{2}-4x+4\right)=\left(x-7\right)\left(x-3\right)\left(x-2\right)\left(x-5\right)-\left(x-6\right)\left(x-3\right)\left(x-2\right)\left(x-6\right)$$

Use the distributive property to multiply x-7 by x-6 and combine like terms.

$$x^{4}-17x^{3}+97x^{2}-207x+126-\left(x^{2}-13x+42\right)\left(x^{2}-4x+4\right)=\left(x-7\right)\left(x-3\right)\left(x-2\right)\left(x-5\right)-\left(x-6\right)\left(x-3\right)\left(x-2\right)\left(x-6\right)$$

Use the distributive property to multiply x^{2}-13x+42 by x^{2}-4x+4 and combine like terms.

$$x^{4}-17x^{3}+97x^{2}-207x+126-\left(x^{4}-17x^{3}+98x^{2}-220x+168\right)=\left(x-7\right)\left(x-3\right)\left(x-2\right)\left(x-5\right)-\left(x-6\right)\left(x-3\right)\left(x-2\right)\left(x-6\right)$$

To find the opposite of x^{4}-17x^{3}+98x^{2}-220x+168, find the opposite of each term.

$$x^{4}-17x^{3}+97x^{2}-207x+126-x^{4}+17x^{3}-98x^{2}+220x-168=\left(x-7\right)\left(x-3\right)\left(x-2\right)\left(x-5\right)-\left(x-6\right)\left(x-3\right)\left(x-2\right)\left(x-6\right)$$

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

$$-17x^{3}+97x^{2}-207x+126+17x^{3}-98x^{2}+220x-168=\left(x-7\right)\left(x-3\right)\left(x-2\right)\left(x-5\right)-\left(x-6\right)\left(x-3\right)\left(x-2\right)\left(x-6\right)$$

Combine -17x^{3} and 17x^{3} to get 0.

$$97x^{2}-207x+126-98x^{2}+220x-168=\left(x-7\right)\left(x-3\right)\left(x-2\right)\left(x-5\right)-\left(x-6\right)\left(x-3\right)\left(x-2\right)\left(x-6\right)$$

Combine 97x^{2} and -98x^{2} to get -x^{2}.

$$-x^{2}-207x+126+220x-168=\left(x-7\right)\left(x-3\right)\left(x-2\right)\left(x-5\right)-\left(x-6\right)\left(x-3\right)\left(x-2\right)\left(x-6\right)$$

Combine -207x and 220x to get 13x.

$$-x^{2}+13x+126-168=\left(x-7\right)\left(x-3\right)\left(x-2\right)\left(x-5\right)-\left(x-6\right)\left(x-3\right)\left(x-2\right)\left(x-6\right)$$

Subtract 168 from 126 to get -42.

$$-x^{2}+13x-42=\left(x-7\right)\left(x-3\right)\left(x-2\right)\left(x-5\right)-\left(x-6\right)\left(x-3\right)\left(x-2\right)\left(x-6\right)$$

Multiply x-6 and x-6 to get \left(x-6\right)^{2}.

$$-x^{2}+13x-42=\left(x-7\right)\left(x-3\right)\left(x-2\right)\left(x-5\right)-\left(x-6\right)^{2}\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply x-7 by x-3 and combine like terms.

$$-x^{2}+13x-42=\left(x^{2}-10x+21\right)\left(x-2\right)\left(x-5\right)-\left(x-6\right)^{2}\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply x^{2}-10x+21 by x-2 and combine like terms.

$$-x^{2}+13x-42=\left(x^{3}-12x^{2}+41x-42\right)\left(x-5\right)-\left(x-6\right)^{2}\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply x^{3}-12x^{2}+41x-42 by x-5 and combine like terms.

$$-x^{2}+13x-42=x^{4}-17x^{3}+101x^{2}-247x+210-\left(x-6\right)^{2}\left(x-3\right)\left(x-2\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$-x^{2}+13x-42=x^{4}-17x^{3}+101x^{2}-247x+210-\left(x^{2}-12x+36\right)\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply x^{2}-12x+36 by x-3 and combine like terms.

$$-x^{2}+13x-42=x^{4}-17x^{3}+101x^{2}-247x+210-\left(x^{3}-15x^{2}+72x-108\right)\left(x-2\right)$$

Use the distributive property to multiply x^{3}-15x^{2}+72x-108 by x-2 and combine like terms.

$$-x^{2}+13x-42=x^{4}-17x^{3}+101x^{2}-247x+210-\left(x^{4}-17x^{3}+102x^{2}-252x+216\right)$$

To find the opposite of x^{4}-17x^{3}+102x^{2}-252x+216, find the opposite of each term.

$$-x^{2}+13x-42=x^{4}-17x^{3}+101x^{2}-247x+210-x^{4}+17x^{3}-102x^{2}+252x-216$$

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

$$-x^{2}+13x-42=-17x^{3}+101x^{2}-247x+210+17x^{3}-102x^{2}+252x-216$$

Combine -17x^{3} and 17x^{3} to get 0.

$$-x^{2}+13x-42=101x^{2}-247x+210-102x^{2}+252x-216$$

Combine 101x^{2} and -102x^{2} to get -x^{2}.

$$-x^{2}+13x-42=-x^{2}-247x+210+252x-216$$

Combine -247x and 252x to get 5x.

$$-x^{2}+13x-42=-x^{2}+5x+210-216$$

Subtract 216 from 210 to get -6.

$$-x^{2}+13x-42=-x^{2}+5x-6$$

Add x^{2} to both sides.

$$-x^{2}+13x-42+x^{2}=5x-6$$

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

$$13x-42=5x-6$$

Subtract 5x from both sides.

$$13x-42-5x=-6$$

Combine 13x and -5x to get 8x.

$$8x-42=-6$$

Add 42 to both sides.

$$8x=-6+42$$

Add -6 and 42 to get 36.

$$8x=36$$

Divide both sides by 8.

$$x=\frac{36}{8}$$

Reduce the fraction \frac{36}{8} to lowest terms by extracting and canceling out 4.

$$x=\frac{9}{2}$$