Pair of Linear Equations in Two Variables Test

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Pair of Linear Equations in Two Variables Test - 31

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Question 1
1 / -0

Solve the following equations by substitution method.
$$x=2y-1; \, \, y=2x-7$$

A
$$x = 8, y= 3$$

B
$$x = 2, y= 3$$

C
$$x = 5, y= 3$$

D
$$x = 1, y= 3$$

Solution

$$ x=2y-1; \, \, y=2x - 7 $$
$$ x = 2y-1 $$ Put it in second equation.
$$ y = 2 \times (2y -1) -7 $$
$$ y =4y -9 $$
$$ \boxed{y = 3} $$

and
$$ x =2y -1 = 2 \times 3 -1 =5 $$
$$\boxed{x=5}$$
$$ \boxed{\,x = 5 \ and \ y = 3 \,}$$
Question 2
1 / -0

Solve the following equations by substitution method.
$$3a-2b=-10; \, \, 2a+3b=2$$

A
$$a = -2, b= 2$$

B
$$a = -1, b= 2$$

C
$$a= -4, b= 2$$

D
$$a = -7, b= 2$$

Solution

$$ 3a -2b =-10; \, \, 2a +3b =2 $$

$$ a = \dfrac {2b-10}{3} \Rightarrow $$ Put it in second equation.

$$ 2 \times \dfrac {2b-10}{3} +3b=2 $$

$$ 4b - 20 +9b = 6 $$
$$ 13b =26 $$
$$ \therefore b=2 $$

$$ a = \dfrac {2b-10}{3} = \dfrac {2(2)-10}{3} = -2 $$

$$ a = -2$$ and $$b = 2 $$
Question 3
1 / -0

Solve the following equations by substitution method.
$$2x-3y-3=7; \, \, 4x-5y-5=10$$

A
$$x = \displaystyle \frac{-7}{2}, y= -5$$

B
$$x = \displaystyle \frac{-3}{2}, y= -5$$

C
$$x = \displaystyle \frac{-1}{2}, y= -5$$

D
$$x = \displaystyle \frac{-5}{2}, y= -5$$

Solution

Rewriting the given equations, we get
$$2x-3y=10$$                              ...(i)
$$4x-5y=15$$                           ...(ii)
We pick either of the equations and write one variable in terms of the other. Consider the Equation (i) and write it as $$x=\displaystyle \frac {10+3y}{2}$$       ...(iii)
Substitute the value of $$x=\displaystyle \frac {10+3y}{2}$$ in Equation (ii). we get,
$$ 4\left(\displaystyle \dfrac {10+3y}{2}\right)-5y=15$$
$$ \Rightarrow 20+6y-5y=15$$
$$\Rightarrow y=-5 $$
Substituting this value of $$y=-5$$ in Equation (iii). we get,
$$x=\displaystyle \frac {10+3(-5)}{2}= \frac {-5}{2}$$

$$\therefore x=\dfrac {-5}{2} , y=-5$$
Question 4
1 / -0

Solve the following simultaneous equation.
$$12x+17y=53; \, \, 17x+12y=63$$

A
$$x = 3,\, y=1$$

B
$$x = 2,\, y=1$$

C
$$x = 1,\, y=1$$

D
$$x = 9,\, y=1$$

Solution

$$12x+17y=53$$                              ...(i)
$$17x+12y=63$$                           ...(ii)
On adding (i) and (ii) , we get
$$29x+29y=116$$
$$\Rightarrow x+y=4$$               ....(iii)
On subtracting (i) from (ii) , we get
$$5x-5y=10$$
$$\Rightarrow x-y=2$$              ....(iv)
On adding (iii) and (iv) , we get
$$2x=6$$
$$\Rightarrow x=3$$
By putting $$x=3$$ in (iii), we get
$$3+y=4$$
$$\Rightarrow y=1$$
Question 5
1 / -0

The ratio of the present ages of mother and son is $$ 12: 5$$. The mother's age at the time of the birth of the son was $$21$$ years. Find their present ages.

A
mother age $$=$$ $$36$$ years, son age $$=$$ $$15$$ years

B
mother age $$=$$ $$56$$ years, son age $$=$$ $$5$$ years

C
mother age $$=$$ $$46$$ years, son age $$=$$ $$25$$ years

D
mother age $$=$$ $$47$$ years, son age $$=$$ $$26$$ years

Solution

Let mother age be $$M$$ and that of son age be $$S$$

Therefore according to question,
$$M = S+21$$ ...(1)
and $$ 5M =12S$$ ....(2)

Substitute equation (1) in equation (2), we get
$$5\left(S+21\right) = 12S$$
$$\Rightarrow 5S+105 = 12S$$
$$\Rightarrow 7S= 105$$
$$\Rightarrow S= 15$$

Using equation (1), we get
$$M= 15+21 = 36$$

Mother age is $$36$$ years and Son age is $$15$$ years.
Question 6
1 / -0

If $$\displaystyle \frac{x+y-8}{2} = \frac{x+2y-14}{3}=\frac{3x-y}{4}$$, then the values of $$x$$ and $$y$$ is

A
$$x=1, \, y=3$$

B
$$x=5, \, y=2$$

C
$$x=3, \, y=3$$

D
$$x=2, \, y=6$$

Solution

Given, $$\displaystyle \frac{x+y-8}{2} = \frac{x+2y-14}{3}=\frac{3x-y}{4}$$

By taking first two. we get,

$$\displaystyle \frac{x+y-8}{2} = \frac{x+2y-14}{3}$$

$$\Rightarrow 3x+3y-24=2x+4y-28$$
$$\Rightarrow x-y=-4$$ ......(i)

By taking last two. we get,

$$\displaystyle \frac{x+2y-14}{3}=\frac{3x-y}{4}$$

$$\Rightarrow 4x+8y-56=9x-3y$$
$$\Rightarrow 5x-11y=-56$$ .....(ii)

On multiplying equation (i) by $$5$$. we get,
$$5x-5y=-20$$ ....(iii)

On subtracting (ii) from (iii). we get,
$$5x-5y-(5x-11y)=-20-(-56)$$
$$\therefore 6y=36$$
$$\therefore y=6$$

Substitute $$y=6$$ in (i). we get,
$$x-6=-4$$
$$\therefore x=2$$

$$\therefore x=2 , y=6$$
Question 7
1 / -0

A two-digit number is 3 more than six times the sum of its digits. If 18 is added to the number obtained by interchanged by interchanging the digits, we get the original number. Find the number.

A
25

B
45

C
75

D
None of these

Solution

Let the tens and the units digits in the number be x and y, respectively.
So, the number may be written as $$10x+y$$.
According to the given condition.
$$10x+y=6(x+y)+3 \Rightarrow 4x-5y=3$$              .....(i)
If we interchange the digits of Original number then we get new number. i.e $$10y+x$$
According to the given condition
$$ 10y+x+18=10x+y$$
$$\Rightarrow 9x-9y = 18 \Rightarrow x-y=2$$                .....(ii)
Now multiplying equation (ii) by $$4$$. we get,
$$4x-4y=8$$          ....(iii)
On subtracting (iii) from (i). we get,
$$4x-5y-(4x-4y)=3-8$$
$$\therefore y=5$$
On substituting $$y=5$$ in (ii). we get,
$$x-5=2 \Rightarrow x=7$$
$$\therefore$$ number is $$10x+y = 10(7)+5=75$$
Question 8
1 / -0

From the following figure, we can say:
$$\displaystyle \frac{x}{3}+\frac{y}{4}=4; \, \, \frac{5x}{6}-\frac{y}{8}=4 $$

A
$$x=1, \, y=8$$

B
$$x=7, \, y=8$$

C
$$x=6, \, y=8$$

D
$$x=3, \, y=8$$

Solution

Given , $$\displaystyle \frac{x}{3}+\frac{y}{4}=4; \, \, \frac{5x}{6}-\frac{y}{8}=4 $$
Rewriting the given equations, we get
$$4x+3y=48$$                              ...(i)
$$20x-3y=96$$                           ...(ii)
On adding (i) and (ii), we get
$$24x=144$$
$$\Rightarrow x=6$$
On putting $$x=6$$ in (i), we get
$$4(6)+3y=48$$
$$\Rightarrow y=8$$
$$\therefore x=6,y=8$$
Question 9
1 / -0

Point $$A$$ and $$B$$ are $$70\ km$$ apart on a highway. A car starts from $$A$$ and another car starts from $$B$$ at the same time. If they travel in the same direction, they meet in $$7$$ hours, but if they travel towards each other they meet in one hour. What are their speeds?

A
$$30\ km/hr$$ and $$40\ km/hr$$

B
$$36\ km/hr$$ and $$40\ km/hr$$

C
$$19\ km/hr$$ and $$20\ km/hr$$

D
$$40\ km/hr$$ and $$50\ km/hr$$

Solution

Let car speed from point $$A=x\, km/hr$$.
and car speed from point $$B=y\, km/hr$$.

Distance = speed $$\times$$ time
$$\therefore$$ Distance from $$A = x \times 7 = 7x$$
Distance from $$B = y \times 7 = 7y$$
If they travel in the same direction, they meet in $$7$$ hours. Point $$A$$ and $$B$$ are $$70 km$$ apart.
$$\Rightarrow 7x-7y=70$$
$$\Rightarrow x-y=10$$ ...$$(i)$$

if they travel towards each other they meet in one hour. Point $$A$$ and $$B$$ are $$70 km$$ apart.
$$x+y=70$$ ...$$(ii)$$

Adding $$(i)$$ and $$(ii)$$. we get
$$2x=80$$
$$\therefore x=40 $$

substituting $$x=40$$ in $$(i)$$. we get
$$40-y=10$$
$$\therefore y=30$$
$$\therefore x=40 km/hr$$ and $$y=30 km/hr$$.
Question 10
1 / -0

The area of a rectangle gets reduced by $$9$$ sq. units, if its length is reduced by $$5$$ units and the breadth, is increased by $$3$$ units. If we increase the length by $$3$$ units and breadth by $$2$$ units, the area is increased by $$67$$ sq. units. Find the length and breadth of the rectangle.

A
length $$ = 12$$, breadth $$=6$$

B
length $$ = 20$$, breadth $$=15$$

C
length $$ = 17$$, breadth $$=9$$

D
length $$ = 21$$, breadth $$=8$$

Solution

Let the length and breadth of the rectangle be $$x$$ and $$y$$.
$$\therefore$$ Area of rectangle $$=xy$$

According to given condition:
$$(x-5)(y+3)=xy-9$$
$$\therefore xy+3x-5y-15=xy-9 \Rightarrow 3x-5y=6$$ ....$$(i)$$

and
$$(x+3)(y+2)=xy+67$$
$$\therefore xy+2x+3y+6=xy+67 \Rightarrow 2x+3y=61$$ .....$$(ii)$$

On multiplying equation $$(i)$$ by $$2$$, we get,
$$6x-10y=12$$ ....$$(iii)$$

On multiplying equation $$(ii)$$ by $$3$$, we get,
$$6x+9y=183$$ .....$$(iv)$$

On subtracting $$(iv)$$ from $$(iii)$$, we get,
$$-19y=-171 \Rightarrow y=9$$

by putting $$y=9$$ in $$(i)$$. we get,
$$3x-5(9)=6 \Rightarrow x=17$$

$$\therefore$$ length $$=17$$ and breadth $$=9$$

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Pair of Linear Equations in Two Variables Test - 31

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Pair of Linear Equations in Two Variables Test - 31

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Question 1

$$x = 8, y= 3$$

$$x = 2, y= 3$$

$$x = 5, y= 3$$

$$x = 1, y= 3$$

Solution

Question 2

$$a = -2, b= 2$$

$$a = -1, b= 2$$

$$a= -4, b= 2$$

$$a = -7, b= 2$$

Solution

Question 3

$$x = \displaystyle \frac{-7}{2}, y= -5$$

$$x = \displaystyle \frac{-3}{2}, y= -5$$

$$x = \displaystyle \frac{-1}{2}, y= -5$$

$$x = \displaystyle \frac{-5}{2}, y= -5$$

Solution

Question 4

$$x = 3,\, y=1$$

$$x = 2,\, y=1$$

$$x = 1,\, y=1$$

$$x = 9,\, y=1$$

Solution

Question 5

mother age $$=$$ $$36$$ years, son age $$=$$ $$15$$ years

mother age $$=$$ $$56$$ years, son age $$=$$ $$5$$ years

mother age $$=$$ $$46$$ years, son age $$=$$ $$25$$ years

mother age $$=$$ $$47$$ years, son age $$=$$ $$26$$ years

Solution

Question 6

$$x=1, \, y=3$$

$$x=5, \, y=2$$

$$x=3, \, y=3$$

$$x=2, \, y=6$$

Solution

Question 7

25

45

75

None of these

Solution

Question 8

$$x=1, \, y=8$$

$$x=7, \, y=8$$

$$x=6, \, y=8$$

$$x=3, \, y=8$$

Solution

Question 9

$$30\ km/hr$$ and $$40\ km/hr$$

$$36\ km/hr$$ and $$40\ km/hr$$

$$19\ km/hr$$ and $$20\ km/hr$$

$$40\ km/hr$$ and $$50\ km/hr$$

Solution

Question 10

length $$ = 12$$, breadth $$=6$$

length $$ = 20$$, breadth $$=15$$

length $$ = 17$$, breadth $$=9$$

length $$ = 21$$, breadth $$=8$$

Solution

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