NCERT Class 10th Mathematics
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NCERT Class 10th – Chapter 3- Exercise 3.6 Pair of Two Equations in Two Variables

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You can see the solution for complete chapter here –

EXERCISE -3.6

QUE:-  1. Solve the following pairs of equations by reducing them to a pair of linear equations:

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

               (ii) 2/√x  +3/√y =2 ; 4/√x -9/√y =-1

               (iii) 4/x +3y =14 ; 3/x -4y =23

             (iv) 5/x -1 +1/ y-2 = 2 ; 6/x-1 – 3/y-2=1

             (v)( 7x-2y)/xy =5 ; 8x +7y/xy=15

            (vi) 6x +3y =6xy ; 2x +4y =5xy

           (vii) (10/x+y) + (2/x-y )=4 ; (15/x+y)-(5 /x-y)=-2

          (viii) (1/3x +y) +(1/3x -y )=3/4 ; (1/2(3x+y))- 1/2(3x-y) =-1/8

   SOL:-

(i) 1/2x +1/3y  =2 ; 1/3x+1/2y =13/6

let 1/x=p and 1/y  =q

p/2+q/3=2

3p +2q -12=………………(1)

p/3+q/2 =13/6

2p +3q -13=…………………(2)

by cross multipication

p/ -26-(-36)=q/-24-(-39)=1/9-4

p/10 =q/15 =1/5

p/10 =1/5

p =2

q/15 =1/5

q =3

so value of x =1/2

and value of y =1/3

(ii) 2/√x  +3/√y =2 ; 4/√x -9/√y =-1

let 1/√x=p and 1/√y =q

2p +3q -2= 0………….(1)

4p -9q +1=0……………..(2)

by cross multipication

p/ 3-(18)=q/-8-(2)=1/-18-(12)

p/-15 =q/-10=1/-30

p =-15/-30=1/2

q =-10/-30 =1/3

so the 1/√x=p =1/2

√x = 2 

x = 2² =4

and 1/√y =q  =1/3

√y = 3

y  = 3² = 9

   (iii) 4/x +3y =14 ; 3/x -4y =23

let 1/x=p and y  =q

4p +3q -14= 0……….(1)

3p – 4q -23 =0 ………..(2)

by cross multipication

p/ -69-56=q/-42-(-92)=1/-16-9

p/-125 =q/50 =1/-25

p=5 ;q =-2

so x=1/5 and y=-2

 (iv) 5/x -1 +1/ y-2 = 2 ; 6/x-1 – 3/y-2=1

let 1/x-1=p and 1/y-2  =q

5p +q -2=0…………….(1)

6p – 3q -1……………….(2)

by cross multipication

p/ -1-6=q/-12-(-5)=1/-12-(-5)

p/-7 = q/-7 =1/-7

p =1 ,q= 1

1/x-1=1

x = 2

1/y-2 =1

y=3

(v)( 7x-2y)/xy =5 ;

8x +7y/xy=15

7/y -2/x =5……………..(1)

8/y +7/x =15………….(2)

let 1/x=p and 1/y =q

7q -2p =5

8q +7p=15

by cross multipication

p/ -105-(-40)=q/-35-30=1/-16-49

p/-65=q/-65=1/-65

p=-65/-65=1

q=-65/-65=1

so 1/x =p

x =1

1/y =q

y= 1

(vi) 6x +3y =6xy ;

2x +4y= 5xy

6/y +3/x =6 ………………….(1)

2/y +4/x =5…………………..(2)

let 1/x=p and 1/y =q

6q -3p =6 ……………(3)

2q +4p=5…………(4)

multypling equ.(4) by 3

(2q +4p=5) ×3

6q +12q =15 …………..(5)

substracting equ. (5) by equ.(3)

6q +12p =15

     -6q -3p =-6

9p=9

p = 9/9=1

putting the value p in equ. (3)

6q -3 =6

6q =3

q=1/2

p=1/x ; x = 1/p =1/1 =1

q =1/y ; y=1/q =1/1/2=2

(vii) (10/x+y) + (2/x-y )=4 ;

(15/x+y)-(5 /x-y)=-2

let 1/x+y=p and 1/x-y =q

10p +2q -4.=0………..(1)

15p -5q+2 = 0 …………(2)

by cross multipication

p/ 4-20=q/-60-20=1/-50-30

p/-16 =q/-80 =1/-80

p=-16/-80=1/5

q= -80/-80=1

1/x+y=p=1/5

5=x +y

x +y =5…………(3)

1/x-y =q=1

x – y =1………..(4)

adding equ(3) and equ.(4)

x – y =1

+  x +y =5

2x =6

x= 3

putting the value of x in equ(3)

3 +y =5

y = 2

(viii) (1/3x +y) +(1/3x -y )=3/4 ;

(1/2(3x+y))- 1/2(3x-y) =-1/8

let 1/3x +y =p and 1/3x -y =q

p +q = 3/4 …………(1)

p/2 -q/2 =-1/8

p -q  =-1/4 ……………..(2)

adding equ. (1) and  equ(2)

p +q = 3/4

    p -q  =-1/4

2p  = 2/4

p =1/4

putting the value of p in equ (1)

p +q = 3/4

1/4 +q = 3/4

q = 2/4=1/2

1/3x +y =p =1/4

3x +y = 4 ………………(3)

1/3x -y =q =1/2

3x -y =2………………..(4)

adding the equ. (3) equ.(4)

3x +y = 4

   3x -y =2

6x =6

x =1

putting the value of x in equ. (3)

3+y = 4

y = 1

so the value of x =1 and value y = 1

QUE:-2. Formulate the following problems as a pair of equations, and hence find their solutions:
          (i) Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her
           speed of rowing in still water and the speed of the current.
       (ii) 2 women and 5 men can together finish an embroidery work in 4 days, while 3
           women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to
          finish the work, and also that taken by 1 man alone.
     (iii) Roohi travels 300 km to her home partly by train and partly by bus. She takes 4
          hours if she travels 60 km by train and the remaining by bus. If she travels 100 km
        by train and the remaining by bus, she takes 10 minutes longer. Find the speed of
        the train and the bus separately.

SOL:-

(i) let the speed of Ritu in still water is x km/h

and speed of steam is y km/h

according to the que

downstreem x + y

2( x +y) =20 ;x +y =10…………(1)

in upstreem x-y

2( x -y) =4

x – y =2 ………………(2)

adding equ. (1) and (2)

x +y =10

    x – y =2

2x  = 12

x = 12/2 =6

putting the value of x in equ.(1)

6+y =10

y  = 4

so the speed of Ritu in still water is 6 km/h

and speed of steam is 4 km/h

(ii) let the work completed by 1 man in x days

and the work completed by 1 woman in y days

work done by man in one day 1/x

work done by woman in one day 1/y

according to given information

4(5/x +2/y)=1

5/x +2/y =1/4

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

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

let 1/x =p and 1/y = q

5p +2q =1/4

20p +8q =1 ………………(1)

6p +3q=1/3

18p +9q =1………………(2)

by cross multipication

p/ -20-(-18)=q/-9-(-8)=1/144-180

p /-2 =q/-1=1/-36

p = -2/-36=1/18

q =-1/-36=1/36

1/x =p =1/18

x =18

1/y = q

y =36

so the work completed by 1 man in 18 days

and the work completed by 1 woman in 36days

(iii) let the speed of train is x km/h

and the speed of bus is ykm/h

acoording to the question

240/x +60/y =4……………….(1)

and 200/x +100/y =25/6……………(2)

let 1/x =p and 1/y =q

240p +60q =4 …………….(3)

200p +100q =25/6

1200p + 600q =25 …………(4)

by cross multipication

p/ -1500-(-2400)=q/-4800-(-6000)=1/144000-72000

p/900=q/1200 =1/72000

p/900=1/72000

p =1/80

q=1/60

so x =80 and y =60

the speed of train is 80 km/h

and the speed of bus is60 km/h

You can see the solution for complete chapter here –

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