Class 10 Maths

# Linear Equations

## NCERT Exercise 3.6

### Part 1

Question 1: Solve the following pairs of equations by reducing them to a pair of linear equations:

(a) (1)/(2x)+(1)/(3y)=2 and (1)/(3x)+(1)/(2y)=(13)/(6)

(1)/(2x)+(1)/(3y)=2

Or, (3y+2x)/(6xy)=2

Or, 2x+3y=12xy

Second equation:

(1)/(3x)+(1)/(2y)=(13)/(6)

Or, (2y+3x)/(6xy)=(13)/(6)

Or, 6(2y+3x)=12xx6xy
Or, 18x+12y=78xy

Multiplying the first equation by 4 and subtracting the resultant from second equation, we get;

18x + 12y – 8x – 12y = 78xy – 48xy
Or, 10x = 30xy
Or, 1 = 3y
Or, y = 1/3

Substituting the value of x in first equation, we get;

2x+3y=12xy
Or, 2x+3xx1/3=2x\xx1/3

Or, 2x+1=4x
Or, 2x=1
Or, x=1/2

Hence, x=1/2 and y=1/3

(b) (2)/(sqrtx)+(3)/(sqrty)=2 and (4)/(sqrtx)-(9)/(sqrty)=-1

(2)/(sqrtx)+(3)/(sqrty)=2

Or, (3sqrtx+2sqrty)/(sqrt(xy))=2

Or, 3sqrtx+2sqrty=2sqrt(xy)

Second equation:

(4)/(sqrtx)-(9)/(sqrty)=-1

Or, (4sqrty-9sqrtx)/(sqrt(xy))=-1

Or, 4sqrty-9sqrtx=-sqrt(xy)
Or, 9sqrtx-4sqrty=sqrt(xy)

From first and second equation, it is clear;

3sqrtx+2sqrty=2(9sqrtx-4sqrty)
Or, 3sqrtx+2sqrty=18sqrtx-8sqrty
Or, 3sqrtx+2sqrty+8sqrty=18sqrtx
Or, 10sqrty=15sqrtx
Or, sqrtx=(10)/(15)sqrty=(2)/(3)sqrty

Substituting the value of √x in any of the above equation, we get;

3sqrtx+2sqrty=2sqrt(xy)
Or, 3xx(2)/(3)sqrty+2sqrty=2xx(2)/(3)sqrty\xx\sqrty

Or, 2sqrty+2sqrty=(4)/(3)y

Or, 4sqrty=(4)/(3)y

Or, sqrty=4÷(4)/(3)=3

Substituting the value of √y in second equation, we get;

9sqrtx-4sqrty=sqrt(xy)
Or, 9sqrtx-4xx3=3sqrtx
Or, 9sqrtx-3sqrtx=12
Or, 6sqrtx=12
Or, sqrtx=2
Hence, x=4 and y=9

(c) (4)/(x)+3y=14 and (3)/(x)-4y=23

(4)/(x)+3y=14

Or, (4+3xy)/(x)=14

Or, 4+3xy=14x
Or, 4+3xy-14x=0

Second equation:

(3)/(x)-4y=23

Or, (3-4xy)/(x)=23

Or, 3-4xy=23x
Or, 3-4xy-23x=0

Multiply first equation by 4 and second equation by 3 and add the resultant equations;

16 + 12xy – 56x + 9 - 12xy - 69x = 0
Or, 25 – 125x = 0
Or, 125x = 25
Or, x = 1/5

Substituting the value of x in first equation, we get;

4+3xy-14x=0
Or, 4+3xx(1)/(5)y-14xx(1)/(5)=0

Or, (20+3y-14)/(5)=0

Or, 6+3y=0
Or, 3y=-6
Or, y=-(6)/(3)=-2

Here, x=1/5 and y=-2