10th Maths 3.6

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. :

$1/(2x)+1/(3y)=2$ ___________(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 :

$x=4, y=5$

$(7x-2y)/(xy)=5$

$7/y-2/x=5$ __________(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. :

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.

Comments

Unknown said…

Nice 👍

26 November 2022 at 21:45

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