Pair of Linear Equations in Two Variables Test

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Pair of Linear Equations in Two Variables Test - 59

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Weekly Quiz Competition

Question 1
1 / -0

Solve each pair of equation by using the substitution method.
$$0.2x+0.3{y}=1.3$$
$$0.4x+0.5{y}=2.3$$

A
$$x=-3$$ and $$y=-2$$

B
$$x=2$$ and $$y=2$$

C
$$x=3$$ and $$y=2$$

D
None of these

Solution

$$0.2x+0.3y=1.3.......(i)$$
$$0.4x+0.5y=2.3......(ii)$$
From equation $$(i)$$, we get
$$0.2x=1.3-0.3y$$
$$\Rightarrow \ x=\dfrac {13-3y}{2}.....(iii)$$
From equation $$(iii)$$ and equation $$(ii)$$
$$\therefore \ 0.4\left(\dfrac {13-3y}{2}\right)+0.5y=2.3$$
$$\Rightarrow \ 2.6-0.6y+0.5y=2.3$$
$$\Rightarrow \ -0.1y=-0.3\\\ \Rightarrow y=3$$
$$\therefore x=\dfrac {13-3\times 3}{2}\\\ \ \ \ \ \ =2$$
Question 2
1 / -0

Solve each pair of equation by using the substitution method.
$$2x+3y=9$$
$$3x+4y=5$$

A
$$x=-21$$ and $$y=7$$

B
$$x=1$$ and $$y=17$$

C
$$x=-21$$ and $$y=17$$

D
None of these

Solution

$$2x+3y=9---(i)$$
$$3x+4y=5--(ii)$$
From equation $$(i)$$, we get
$$\Rightarrow \ 2x=9-3y\ \Rightarrow x=\dfrac {9-3y}{2}---(iii)$$
Substituting equation $$(iii)$$ in equartion $$(ii)$$, we get
$$\Rightarrow \ 3\left(\dfrac {9-3y}{2}\right)+4y=5$$
$$\Rightarrow \ 27-9y+8y=10$$
$$\Rightarrow \ y=27-10=17$$
$$\therefore \ x=\dfrac {9-3\times 17}{2}=\dfrac {-42}{2}=-21$$
Question 3
1 / -0

Classes A and B have $$35$$ students each. If seven girls shift from class $$A$$ to class $$B$$, then the number of girls in the classes would interchange. If four girls shift from class $$B$$ to class $$A$$, then the number of girls in class $$A$$ would be twice the original number of girls in class $$B$$. What is the number of boys in Class $$A$$ and in Class $$B$$?

A
$$18$$ and $$11$$

B
$$24$$ and $$19$$

C
$$18$$ and $$27$$

D
$$17$$ and $$24$$

Solution

$$\textbf{Step -1: Form the required equations.}$$

$$\text{Let the number of girls in class }A\text{ be }x.$$

$$\text{Let the number of girls in class }B\text{ be }y.$$

$$\text{Let the number of boys in class }A\text{ be }35-x.$$

$$\text{Let the number of boys in class }B\text{ be }35-y.$$

$$\text{If seven girls are shifted from class }A\text{ to }B,$$

$$\text{Number of girls in class }A\text{ would become }x-7.$$

$$\text{Number of girls in class }B\text{ would become }y+7.$$

$$\text{But according to the question,}$$

$$y=x-7$$

$$\Rightarrow x-y=7\ldots(i)$$

$$\text{If four girls are shifted from class }B\text{ to }A,$$

$$\text{Number of girls in class }A\text{ would become }x+4.$$

$$\text{Number of girls in class }B\text{ would become }y-4.$$

$$\text{But according to the question,}$$

$$x+4=2y$$

$$\Rightarrow x-2y=-4\ldots(ii)$$

$$\textbf{Step -2: Solve the above formed two equations simultaneously.}$$

$$\text{On subtracting equation }(ii)\text{ from }(i),\text{ we get}$$

$$-y+2y=7+4$$

$$\Rightarrow y=11$$

$$\text{On putting the value of }y\text{ in equation }(i),\text{ we get}$$

$$x-11=7$$

$$\Rightarrow x=18$$

$$\textbf{Step -3: Find the number of boys in class A and B.}$$

$$\text{Number of boys in class }A=35-18$$

$$=17$$

$$\text{Number of boys in class }B=35-11$$

$$=24$$

$$\textbf{Therefore, option D is correct.}$$
Question 4
1 / -0

A number is $$\dfrac{2}5{}$$ times another number. If their sum is $$70$$, Find the numbers.

A
$$70$$ and $$50$$

B
$$20$$ and $$30$$

C
$$40$$ and $$50$$

D
$$20$$ and $$50$$

Solution

Let the numbers be $$x$$ and $$ y$$
According to the question
$$x=\dfrac{2}{5}y............(i)$$
$$x+y=70...............(ii)$$
$$\dfrac{2}{5}y+y=70$$
$$\dfrac{2y+5y}{5}=70$$
$$7y=350$$
$$y=50$$
$$x+50=70$$
$$x=20$$
So the numbers are $$20$$ and $$50$$
Question 5
1 / -0

Solve the following pairs of linear equations by elimination method
$$3x+4y=25$$
$$5x-6y=-9$$

A
$$x=3$$ and $$y=3$$

B
$$x=4$$ and $$y=4$$

C
$$x=3$$ and $$y=4$$

D
None of these

Solution

$$3x+4y=25\_\_\_\_\_\_(i)\times 3$$
$$5x-6y=-9\_\_\_\_\_\_(ii)\times 2$$

$$9x+12y=75$$
$$10x-12y=-18$$
(+)$$\quad$$(+)$$\quad\quad\quad$$(+)
$$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$$
$$19x\quad=57$$
$$\Rightarrow x=3$$
$$\Rightarrow 3 \times 3+4y=25$$
$$\Rightarrow 4y=16\Rightarrow y=4$$
Question 6
1 / -0

Sum of two numbers is $$407$$. The sum and difference of their LCM and HCF are $$925$$ and $$851$$ respectively. The difference of two numbers is

A
$$518$$

B
$$185$$

C
$$158$$

D
$$175$$

Solution

Let the two numbers: be $$x$$ & $$y$$
$$\therefore x+y=407$$ ......... $$(1)$$

Acc to the question,
$$LCM+HCF=925$$ .......... $$(2)$$
$$LCM-HCF=851$$ ......... $$(3)$$

On subtracting $$(2)$$ & $$(3)$$
$$LCM+HCF=925\\ -LCM+HCF=-851\\ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \_ \\ \quad \quad \quad \ \ \ \ \ 2HCF=74\\ \quad \quad \quad \ \ \ \ \ \ \ HCF=37$$

$$\therefore LCM -(37)=857$$
$$LCM=857+37$$
$$=888$$

As we know, $$LCM\times HCF=$$ product of two numbers
$$\therefore 888\times 37=x xy$$ ............. $$(4)$$

From $$(1)$$,
$$x=(407-y)$$
$$\therefore 888\times 37 = (407-y)y$$
$$\Rightarrow y^{2}-407y+832856$$
$$(y-111)(y-296)$$
$$\therefore x=296$$ and $$y=111$$
$$\therefore$$ Difference of two numbers $$=296-111$$
$$=185$$
$$\therefore $$ Option $$B$$ is correct.
Question 7
1 / -0

Solve the following pair of linear equations in two variables (by graph) :
$$2x+3y=5,\ x+6y=25$$

A
$$(1,2)$$

B
$$(1,3)$$

C
$$(1,6)$$

D
None of these
Question 8
1 / -0

The liquids $$X$$ and $$Y$$ are mixed in the ratio of $$3:2$$ and the mixture is sold at $$Rs\;11$$ per liter at a profit of $$10\%$$. If the liquid $$X$$ costs $$Rs\;2$$ more per liter than $$Y$$, the cost of $$X$$ per liter is (in Rs.):

A
$$9.50$$

B
$$10.80$$

C
$$11.75$$

D
$$11$$

Solution

Let Cost of $$X$$ be $$x$$
Cost of $$Y$$ be $$y$$
and their volumes be $$3K$$ and $$2K$$
Now, $$x=y+2$$
Let original price be $$P$$
So, $$P+10$$% of $$P=11$$
$$P+0.1P=11$$
$$1.1P=11$$
$$P=10$$
So, $$\dfrac { 3Kx+2Ky }{ 3K+2K } =\dfrac { 3x+2y }{ 5 } =10$$
$$3x+2y=50$$....(1)
and $$x=y+2$$....(2)
Using (2) in (1), we get
$$\Rightarrow 3y+6+2y=50\Rightarrow 5y=44\Rightarrow y=\dfrac { 44 }{ 5 } =8.8$$ Rs.
$$x=10.8$$Rs.
Question 9
1 / -0

At a zoo, there were parrots and rabbits in the same enclouser. Sravanthi counted 30 heads and 100 legs altogether. how many animals of each type were in the enclouser

A
$$9$$ parrots $$21$$ rabbits

B
$$10$$ parrots and $$20$$ rabbits

C
$$20$$ parrots and $$19$$ rabbits

D
$$11$$ rabbits and $$19$$ parrots

Solution
Question 10
1 / -0

IF $$x - 2y = -1$$, $$y=-\dfrac { 1 }{ 2 } $$, then find the value of $$x$$

A
$$-2$$

B
$$-1$$

C
$$0$$

D
$$2$$

Solution

Given:
$$x-2y=-1$$.....................[1]

$$y=-\dfrac12$$..................[2]

Putting $$y=-\dfrac12$$ in equation $$1$$:

$$x-2\times -\dfrac12=-1$$

$$\implies x+1=-1$$

$$\implies x=-2$$

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Pair of Linear Equations in Two Variables Test - 59

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Pair of Linear Equations in Two Variables Test - 59

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Question 1

$$x=-3$$ and $$y=-2$$

$$x=2$$ and $$y=2$$

$$x=3$$ and $$y=2$$

None of these

Solution

Question 2

$$x=-21$$ and $$y=7$$

$$x=1$$ and $$y=17$$

$$x=-21$$ and $$y=17$$

None of these

Solution

Question 3

$$18$$ and $$11$$

$$24$$ and $$19$$

$$18$$ and $$27$$

$$17$$ and $$24$$

Solution

Question 4

$$70$$ and $$50$$

$$20$$ and $$30$$

$$40$$ and $$50$$

$$20$$ and $$50$$

Solution

Question 5

$$x=3$$ and $$y=3$$

$$x=4$$ and $$y=4$$

$$x=3$$ and $$y=4$$

None of these

Solution

Question 6

$$518$$

$$185$$

$$158$$

$$175$$

Solution

Question 7

$$(1,2)$$

$$(1,3)$$

$$(1,6)$$

None of these

Question 8

$$9.50$$

$$10.80$$

$$11.75$$

$$11$$

Solution

Question 9

$$9$$ parrots $$21$$ rabbits

$$10$$ parrots and $$20$$ rabbits

$$20$$ parrots and $$19$$ rabbits

$$11$$ rabbits and $$19$$ parrots

Solution

Question 10

$$-2$$

$$-1$$

$$0$$

$$2$$

Solution

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