Question

If 4/x-1 + 1/y-2=3/2 and 10/x-1 +14/y-2=19/2 then the value of 3(2x-y)equals

Answer 1

$\large\underline{\sf{Solution-}}$

Given equations are

$\rm :\longmapsto\:\dfrac{4}{x - 1} + \dfrac{1}{y - 2} = \dfrac{3}{2} - - - (1)$

and

$\rm :\longmapsto\:\dfrac{10}{x - 1} + \dfrac{14}{y - 2} = \dfrac{19}{2} - - - (2)$

Let assume that

$\red{\rm :\longmapsto\:\dfrac{1}{x - 1} = u - - - (3)}$

and

$\red{\rm :\longmapsto\:\dfrac{1}{y - 2} = v - - - (4)}$

So, equation (1) and (2) reduces to

$\rm :\longmapsto\:4u + v = \dfrac{3}{2}$

$\rm :\longmapsto\:8u + 2v = 3 - - - (5)$

and

$\rm :\longmapsto\:10u + 14v = \dfrac{19}{2}$

$\rm :\longmapsto\:20u + 28v = 19 - - - (6)$

On multiply equation (5) by 14 we get

$\rm :\longmapsto\:112u + 28v = 42 - - - (7)$

On Subtracting equation (6) from equation (7), we get

$\rm :\longmapsto\:92u = 23$

$\bf\implies \: u= \dfrac{1}{4} - - - (8)$

On substituting the value of u in equation (5), we get

$\rm :\longmapsto\:4 \times \dfrac{1}{4} + v = \dfrac{3}{2}$

$\rm :\longmapsto\:1 + v = \dfrac{3}{2}$

$\rm :\longmapsto\: v = \dfrac{3}{2} - 1$

$\rm :\longmapsto\: v = \dfrac{3 - 2}{2}$

$\bf\implies \: v = \dfrac{1}{2} - - - (9)$

Now, Substitute the value of u and v in equation (3) and (4), we get

$\rm :\longmapsto\:\dfrac{1}{x - 1} = \dfrac{1}{4} \: \: and \: \: \dfrac{1}{y - 2} = \dfrac{1}{2}$

$\rm :\longmapsto\:x - 1 = 4 \: \: and \: \: y - 2 = 2$

$\bf\implies \:x = 5 \: \: and \: \: y = 4$

Now,

Consider,

$\red{\bf :\longmapsto\:3(2x - y)}$

$\rm \: \: = \: 3(2 \times 5 - 4)$

$\rm \: \: = \: 3(10 - 4)$

$\rm \: \: = \: 3 \times 6$

$\rm \: \: = \: 18$

Hence,

$\red{\bf :\longmapsto\:3(2x - y) = 18}$

Concept Used :-

There are 4 methods to solve this type of pair of linear equations.

1. Method of Substitution

2. Method of Eliminations

3. Method of Cross Multiplication

4. Graphical Method

We prefer here Method of Eliminations.

To solve systems using elimination, follow this procedure:

The Elimination Method

Step 1: Multiply each equation by a suitable number so that the two equations have the same leading coefficient.

Step 2: Subtract the second equation from the first to eliminate one variable

Step 3: Solve this new equation for other variable.

Step 4: Substitute the value of variable thus evaluated into either Equation 1 or Equation 2 and get the value other variable.