Question

Solve for x:

$\frac{x - 4}{x - 5} + \frac{x - 6}{x - 7} = \frac{10}{3}$

Class 10

Ans: x = 8 , 11 / 2

Quadratic Equation

Answer 1

$\mathf{ \dfrac{x - 4}{x - 5} + \dfrac{x - 6}{x - 7} = \dfrac{10}{3}$

Make the denominator the same:

$\mathf{ \dfrac{(x - 4)(x-7)}{(x - 5)(x-7)} + \dfrac{(x - 6)(x-5)}{(x - 7)(x-5)} = \dfrac{10}{3} }$

Combine into single fraction:

$\mathf{ \dfrac{(x - 4)(x-7 )+ (x-6)(x-5)}{(x - 5)(x-7)} = \dfrac{10}{3} }$

Expand the expressions:

$\mathf{ \dfrac{x^2-11x+28-x^2-11x+30}{(x - 5)(x-7)} = \dfrac{10}{3} }$

Combine like terms:

$\mathf{ \dfrac{x^2-22x+58}{x^2-12x+35} = \dfrac{10}{3} }$

Cross multiply:

$\mathf{ 3(2x^2-22x+58) = 10(x^2-12x+35)}$

Distribute 3 and 10:

$\mathf{ 6x^2 - 66x + 174 = 10x^2 - 120x + 350}$

Move all the LHS:

$\mathf{4x^2 - 54x + 176 = 0 }$

Factorise:

$\mathf{ 2(x - 8)(2x - 11) = 0}$

Divide both sides by 2:

$\mathf{ (x - 8)(2x - 11) = 0}$

Apply zero product property:

$\mathf{ x - 8 = 0 \: or \: x = \dfrac{11}{2} }$

Answer 2

DEAR STUDENT,

Kindly refer the attached papers for a detailed explanation and step by step solution along with elaborations to reach our final conclusion or our desired or required answer that is,

$\boxed{\bf{\underline{x = 8}}}$

AND,

$\boxed{\bf{\underline{x = \dfrac{11}{2}}}}$

Which are the required solutions for this type of query.

Hope this helps you and clears your doubts for obtaining the variable value of "x" by factoring the terms!!!!