Question

Solve for x and y ,
(x-1 / x-2) + ( x-3 / x - 4 ) = 10/3.

Answer 1

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For solving for x, just take the LCM i.e.

• (x - 2)(x -4)(x−2)(x−4)

Then,

• (x−2)(x−4)(x−1)(x−4)+(x−3)(x−2)= 310(x−2)(x−4)(x−1)(x−4)+(x−3)(x−2)=310

Cross multiplying for easier calculation

• 3(x−1)(x−4)+(x−3)(x−2) = 10(x−2)(−4)

Multiplying the terms inside the curly brackets

• 3x² −5x + 4+x² −5x +6=10(x² −6x+8)

Opening the parentheses,

• 3(2x²−10x+10)=10x² −60x+80
• 6x²−30x+30 =10x²−60x+80

Taking 2 as a common from both sides & cancelling it, we have,

• 3x²−15x+15=5x²−30x+40

Now shifting the terms to one side of the equation:

• 5x²−3x²−30x+15x+40−15=0
• 2x²−15x+25=0

Now find the possible values of x by Middle term factorisation,

• 2x²−10x−5x+25=0
• 2x(x−5)−5(x−5)=0
• (2x−5)(x−5)=0

Equating to 0, We get

• x = 5/2
• x = 5

Hence

• The value of x after solving: 5/2 or 5

Answer 2

Answer:

As given,

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

L.C.M. of (x-2) & (x-4) = (x-2)(x-4)

Then,

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

=> 3(x^2 -5x +4 + x^2 -5x +6) = 10(x-2)(x-4)

=> 3( 2x^2–10x+10) = 10(x^2–6x+8)

=> 6x^2–30x +30 = 10x^2–60x+80

Taking 2 as a common from both sides & cancelling it, we have,

=> 3x^2–15x+15 = 5x^2–30x+40

=>5x^2–3x^2–30x+15x+40–15=0

=> 2x^2 -15x+25=0

Now, by the middle term splitting method,we have,

=> 2x^2 -10x-5x+25=0

=> 2x(x–5) -5(x-5) =0

=> (2x-5)(x-5)=0

=> 2x-5 = 0

So, x= 5/2

Or, x-5= 0

So, x=5

Hence, x=5 or 5/2 Ans

Explanation:

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