Question

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Answer 1

$\frac{x-2}{x-3} + \frac{x-4}{x-5} = \frac{10}{3}$

LCM = 3(x - 3)(x - 5).

$\frac{x - 2}{x - 3} * 3(x - 3)(x - 5)+ \frac{x-4}{x-5} * 3(x - 3)(x - 5) = \frac{10}{3} * 3(x-3)(x-5)$

3(x - 2)(x - 5) + 3(x - 4)(x - 3) = 10(x - 3)(x - 5)

3(x^2 - 5x - 2x + 10) + 3(x^2 - 3x - 4x + 12) = 10(x^2 - 5x - 3x + 15)

3(x^2 - 7x + 10) + 3(x^2 - 7x + 12) = 10(x^2 - 8x + 15)

3x^2 - 21x + 30 + 3x^2 - 21x + 36 = 10x^2 - 80x + 150

6x^2 - 42x + 66 = 10x^2 - 80x + 150

6x^2 - 42x + 66 - 150 = 10x^2 - 80x

6x^2 - 42x - 84 = 10x^2 - 80x

6x^2 - 42x - 84 + 80x = 10x^2

6x^2 + 38x - 84 = 10x^2

-4x^2 + 38x - 84 = 0

-2(2x^2 - 19x - 42) = 0

2x^2 - 19x + 42 = 0

2x^2 - 12x - 7x + 42 = 0

2x(x - 6) - 7(x - 6) = 0

(2x - 7)(x - 6) = 0

x = 7/2 (or) x = 6.


Hope this helps!

Answer 2

ANSWER
..............

LCM = 3(x - 3)(x - 5).

\{x - 2}{x - 3} * 3(x - 3)
(x - 5)+ \{x-4}{x-5} * 3(x - 3)(x - 5)
= \{10}{3} * 3(x-3)(x-5)
​x−3
​
​x−2
​​ ∗3(x−3)(x−5)+
​x−5
​
​x−4
​​ ∗3(x−3)(x−5)=
​3
​
​10
​​ ∗3(x−3)(x−5)

3(x - 2)(x - 5) + 3(x - 4)(x - 3) = 10(x - 3)(x - 5)

3(x^2 - 5x - 2x + 10) + 3(x^2 - 3x - 4x + 12) = 10(x^2 - 5x - 3x + 15)

3(x^2 - 7x + 10) + 3(x^2 - 7x + 12) = 10(x^2 - 8x + 15)

3x^2 - 21x + 30 + 3x^2 - 21x + 36 = 10x^2 - 80x + 150

6x^2 - 42x + 66 = 10x^2 - 80x + 150

6x^2 - 42x + 66 - 150 = 10x^2 - 80x

6x^2 - 42x - 84 = 10x^2 - 80x

6x^2 - 42x - 84 + 80x = 10x^2

6x^2 + 38x - 84 = 10x^2

-4x^2 + 38x - 84 = 0

-2(2x^2 - 19x - 42) = 0

2x^2 - 19x + 42 = 0

2x^2 - 12x - 7x + 42 = 0

2x(x - 6) - 7(x - 6) = 0

(2x - 7)(x - 6) = 0

x = 7/2 (or) x = 6.