Solve for x: (x - 1) / (x - 2) + ( x - 3) / (x - 4) = 10/3

An equation can be defined as a mathematical statement consisting of an equal to symbol between two algebraic expressions that have the same value. 

Answer: The values of x for the equation (x - 1) / (x - 2) + ( x - 3) / (x - 4) = 10/3 are 5/2 and 5.

Let's calculate the value of x for the given equation (x - 1) / (x - 2) + ( x - 3) / (x - 4) = 10/3.

Explanation: 

(x - 1) / (x - 2) + ( x - 3) / (x - 4) = 10/3 

Take the LCM of the denominators of LHS (x-2) & (x-4) which is equal to (x-2)(x-4).

⇒ [(x-1)(x-4)+(x-3)(x-2)]/(x-2)(x-4) = 10/3

⇒ 3(x² - 5x + 4 + x² - 5x + 6) = 10(x-2)(x-4) (By cross multiplying and solving)

⇒ 3(2x² - 10x + 10) = 10(x² -6x + 8)

⇒ 6x² - 30x + 30 = 10x² -60x + 80

⇒ 3x² - 15x + 15 = 5x² -30x + 40 (By dividing the whole equation by 2)

⇒ 5x² - 3x² - 30x + 15x + 40 - 15 = 0

⇒ 2x² - 15x + 25 = 0

By splitting the middle term, we have,

⇒ 2x² - 10x - 5x + 25 = 0

⇒ 2x(x - 5) - 5(x - 5) = 0

⇒ (2x - 5)(x - 5) = 0

⇒ 2x - 5 = 0 or x - 5 = 0

Hence, x = 5/2 or x = 5.

Thus, the values of x for the equation (x - 1) / (x - 2) + ( x - 3) / (x - 4) = 10/3 are 5/2 and 5.