10th Maths 3.6
NCERT Class 10th solution of Exercise 3.1
NCERT Class 10th solution of Exercise 3.5
Exercise 3.6
Q1. 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/sqrtx+3/sqrty=2 ; 4/sqrtx-9/sqrty=-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)
`1/(3x)+1/(2y)=13/6`_________(2)
Let `1/x=p` and `1/y=q` then
`p/2+q/3=2`
`3p+2q=12`________________(3)
`p/3+q/2=13/6`
`2p+3q=13`________________(4)
from equation (3)
`q=(12-3p)/2`
Put in equation (4)
`2p+3((12-3p)/2)=13`
`4p+36-9p=26`
`36-26=5p`
`10=5p`
`p=10/5`
`p=2`
Put in equation (3)
`3times2+2q=12`
`6+2q=12`
`2q=12-6`
`2q=6`
`q=6/2`
`q=3`
Now
`p=1/x`
`2=1/x`
`x=1/2`
and
`q=1/y`
`3=1/y`
`y=1/3`
Answer :
`x=1/2, y=1/3`
ii)
`2/sqrtx+3/sqrty=2`____________(1)
`4/sqrtx-9/sqrty=-1`____________(2)
Let `1/sqrtx=p` and `1/sqrty=q` then
`2p+3q=2`___________________(3)
`4p-9q=-1`___________________(4)
from equation (3)
`p=(2-3q)/2`
Put in equation (4)
`4((2-3q)/2)-9q=-1`
`4-6q-9q=-1`
`-15q=-1-4`
`q=(-5)/(-15)`
`q=1/3`
Put in equation (3)
`4p-9(1/3)=-1`
`4p-3=-1`
`p=(-1+3)/4`
`p=2/4`
`p=1/2`
Now
`p=1/2`
`1/sqrtx=1/2`
`sqrtx=2`
`x=4`
`q=1/3`
`1/sqrty=1/3`
`sqrty=3`
`y=9`
Answer :
`x=4, y=9`
iii)
`4/x+3y=14`____________(1)
`3/x-4y=23`____________(2)
Let `1/x=p`,
then
`4p+3y=14`_____________(3)
`3p-4y=23`_____________(4)
Multiply equation (3) by `4` and equation (4) by `3`
`16p+12y=56`___________(5)
`9p-12y=69`____________(6)
By equation (5) + (6)
`16p+12y+9p-12y=56+69`
`25p=125`
`p=125/25`
`p=5`
Put in equation (3)
`4(5)+3y=14`
`20+3y=14`
`3y=-20+14`
`y=-6/3`
`y=-2`
Now
`p=5`
`1/x=5`
`x=1/5`
Answer :
`x=1/5, y=-2`
iv)
`5/(x-1)+1/(y-2)=2`_________(1)
`6/(x-1)-3/(y-2)=1`__________(2)
Let `1/(x-1)=p` and `1/(y-2)=q` then
`5p+q=2`_________________(3)
`6p-3q=1`________________(4)
from equation (3)
`q=2-5p`
Put in equation (4)
`6p-3(2-5p)=1`
`6p-6+15p=1`
`21p=1+6`
`p=7/21`
`p=1/3`
Put in equation (3)
`5(1/3)+q=2`
`5/3+q=2`
`q=2-5/3`
`q=1/3`
Now
`p=1/3`
`1/(x-1)=1/3`
`x-1=3`
`x=4`
`q=1/3`
`1/(y-2)=1/3`
`y-2=3`
`y=3+2`
`y=5`
Answer :
`x=4, y=5`
v)
`(7x-2y)/(xy)=5`
`7/y-2/x=5`__________(1)
`(8x+7y)/(xy)=15`
`8/y+7/x=15`_________(2)
Let `1/y=p` and `1/x=q` then
`7p-2q=5`___________(3)
`8p+7q=15`__________(4)
from equation (3)
`q=(7p-5)/2`
Put in equation (4)
`8p+7((7p-5)/2)=15`
`16p+49p-35=30`
`65p=30+35`
`p=65/65`
`p=1`
Put in equation (3)
`7(1)-2q=5`
`7-2q=5`
`7-5=2q`
`q=2/2`
`q=1`
Now
`p=1`
`1/y=1`
`y=1`
`q=1`
`1/x=1`
`x=1`
Answer :
`x=1, y=1`
vi)
`6x+3y=6xy`
`6/y+3/x=6`___________(1)
`2x+4y=5xy`
`2/y+4/x=5`___________(2)
Let `1/y=p` and `1/x=q`
`6p+3q=6`____________(3)
`2p+4q=5`____________(4)
Multiply equation (3) by `4` and equation (4) by `3`.
`24p+12q=24`_________(5)
`6p+12q=15`__________(6)
By equation (5) - (6)
`24p+12q-(6p+12q)=24-15`
`24p+12q-6p-12q=9`
`18p=9`
`p=9/18`
`p=1/2`
Put in equation (3)
`6(1/2)+3q=6`
`3+3q=6`
`3q=6-3`
`3q=3`
`q=3/3`
`q=1`
Now
`p=1/2`
`1/y=1/2`
`y=2`
`q=1`
`1/x=1`
`x=1`
Answer :
`x=1, y=2`
vii)
`10/(x+y)+2/(x-y)=4`____________(1)
`15/(x+y)-5/(x-y)=-2`____________(2)
Let `1/(x+y)=p` and `1/(x-y)=q`
`10p+2q=4`___________________(3)
`15p-5q=-2`___________________(4)
Multiply equation (3) by `5` and equation (4) by `2`
`50p+10q=20`________________(5)
`30p-10q=-4`_________________(6)
By equation (5) + (6)
`50p+10q+30p-10q=20-4`
`80p=16`
`p=16/80`
`p=1/5`
Put in equation (3)
`10(1/5)+2q=4`
` 2+2q=4`
`2q=4-2`
`q=2/2`
`q=1`
Now
`p=1/5`
`1/(x+y)=1/5`
`x+y=5`_____________(7)
`q=1`
`1/(x-y)=1`
`x-y=1`_____________(8)
By equation (7) + (8)
`x+y+x-y=5+1`
`2x=6`
`x=3`
Put in (7)
`3+y=5`
`y=2`
Answer :
`x=3, y=2`
viii)
`1/(3x+y)+1/(3x-y)=3/4`___________(1)
`1/(2(3x+y))-1/(2(3x-y))=-1/8`
`1/(3x+y)-1/(3x-y)=-1/4`____________(2)
Let `1/(3x+y)=p` and `1/(3x-y)=q`
`p+q=3/4`_____________________(3)
`p-q=-1/4`_____________________(4)
By equation (3) + (4)
`p+q+p-q=3/4-1/4`
`2p=2/4`
`p=1/4`
Put in equation (3)
`1/4+q=3/4`
`q=3/4-1/4`
`q=(3-1)/4`
`q=2/4`
`q=1/2`
Now
`p=1/4`
`1/(3x+y)=1/4`
`3x+y=4`_____________(5)
`q=1/2`
`1/(3x-y)=1/2`
`3x-y=2`_____________(6)
By equation (5) + (6)
`3x+y+3x-y=4+2`
`6x=6`
`x=1`
Put in equation (5)
`3(1)+y=4`
`y=4-3`
`y=1`
Answer :
`x=1, y=1`
Q2. Formulate the following problems as a pair of equations, and hence find their solutions:
i) Ritu can row downstream `20km` in `2` hours, and upstream `4km` 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 `300km` to her home partly by train and partly by bus. She takes `4` hours if she travels `60km` by train and remaining by bus. If she travels `100km` 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 speed in still water is `x` km/h,
and speed in current is `y` km/h.
According to question
[ Time`times`speed = Distance ]
`2(x+y)=20`
`x+y=10`_____________(1)
[ Time`times`speed=Distance ]
`2(x-y)=4`
`x-y=2`______________(2)
By equation (1) + (2)
`x+y+x-y=10+2`
`2x=12`
`x=6`km/h
Put in equation (1)
`6+y=10`
`y=10-6`
`y=4`km/h
Answer :
Speed in still water is`x=6`km/h, and speed in current is `y=4`km/h.
ii)
Let a woman finish the work in `x` days,
and a man finish the work in `y` days.
One day's work of a woman is `1/x`,
and a man is `1/y`.
According to question
`2/x+5/y=1/4`___________(1)
`3/x+6/y=1/3`___________(2)
Let `1/x=p` and `1/y=q`
`2p+5q=1/4`
`8p+20q=1`___________(3)
`3p+6q=1/3`
`9p+18q=1`___________(4)
`p/(b_1c_2-b_2c_1)``=q/(c_1a_2-c_2a_1)=1/(a_1b_2-a_2b_1)`
`p/((18)(-1)-(-20)(-1))``=q/((-1)(8)-(-1)(9))=1/((9)(20)-(8)(18))`
`p/-(-18+20)=q/(-8+9)=1/(180-144)`
`p/2=q/1=1/36`
`p=2/36=1/18`,
`q=1/36`
Now
`p=1/x`
`1/18=1/x`
`x=18`
`q=1/y`
`1/36=1/y`
`y=36`
Answer :
`x=18` days, `y=36` days.
iii)
Let the speed of train `x`km/h,
and the speed of bus `y`km/h.
When Roohi travels by train `60`km, and by bus `240`km.
Travel time `4`hr.
`60/x+240/y=4`
`15/x+60/y=1`___________(1)
When Roohi travels by train `100`km, and by bus `200`km.
Travel time `4`hr`10`minute`=4+10/60=4+1/6=25/6`hr.
`100/x+200/y=25/6`
`24/x+48/y=1`___________(2)
Let `1/x=p` and `1/y=q`
`15p+60q=1`
`15p+60q-1=0`__________(3)
`24p+48q=1`
`24p+48q-1=0`__________(4)
`p/(b_1c_2-b_2c_1)``=q/(c_1a_2-c_2a_1)=1/(a_1b_2-a_2b_1)`
`p/((60)(-1)-(48)(-1))``=q/((-1)(24)-(-1)(15))=1/((15)(48)-(24)(60))`
`p/(-60+48)=q/(-24+15)=1/(720-1440)`
`p/-12=q/-9=1/-720`
`p=(-12)/-720`
`1/x=1/60`
`x=60`km/h,
`q=(-9)/-720`
`1/y=1/80`
`y=80`km/h.
Answer :
The speed of the train is `60`km/h, and the bus is `80`km/h.
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