Question

`(x-4)/(x-5)+(x-6)/(x-7)=31/3,x≠5,7` 

 

Answer 1

`(x-4)/(x-5)+(x-6)/(x-7)=3 1/3,x≠5,7` 

⇒` ((x-4)(x-7)+(x-5)(x-6))/(x^2-12x+35)=10/3` 

⇒`(x^2-11x+28+x^2-11x+30)/(x^2-12x+35)=10/3` 

⇒`(2x^2-22x+58)/(x^2-12x+35)=10/3` 

⇒` (x^2-22x+29)/(x^2-12x+35)=5/3` 

⇒`(x^2-11x+29)/(x^2-12x+35)=5/3` 

⇒`3x^2-33x+87=5x^2-60x+175` 

⇒`2x^2-27x+88=0` 

⇒`2x^2-16x11x+88=0` 

⇒`2x(x-8)-11(x-8)=0` 

⇒`(x-8) (2x-11)=0` 

⇒`x-8=0  or  2x-11=0` 

⇒`x=8  or  x=11/2` 

Hence, 8 and `11/2` are the roots of the given equation.