Pair of Linear Equations in Two Variables Test

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Pair of Linear Equations in Two Variables Test - 25

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Weekly Quiz Competition

Question 1
1 / -0

Sum of two numbers is 97. If the larger number is divided by the smaller, the quotient is 7 and the remainder is 1. Find the numbers.

A
$$75, 14$$

B
$$90, 14$$

C
$$75, 18$$

D
$$85, 12$$

Solution

$$\textbf{Step-1: Assume the numbers to be equal to some variables& then apply all information's given.}$$

$$\text{Let the two numbers be}$$ $$x$$ $$\text{and}$$ $$y,$$ $$\text{where}$$ $$x$$ $$\text{is the larger number}$$

$$\text{Given, Sum of two numbers is 97}$$

$$ x + y = 97$$ $$\text{...(1)}$$

$$\text{As per the given condition,}$$

$$\text{If the larger number is divided by the smaller, the quotient is 7 and the remainder is 1}$$

$$ x = 7y +1 $$ $$\text{...(2)}$$

$$\textbf{Step-2: Solve above equations.}$$

$$\text{Substitute eq(2) in eq(1) we get,}$$

$$7y+1 + y = 97\\ 8y = 96\\ y = 12\\ \therefore x = 7(12) +1 = 85$$

$$\textbf{Hence, option - D is the answer.}$$
Question 2
1 / -0

Solve the following simultaneous equations :
$$\displaystyle \frac{1}{3x}\, +\, \frac{1}{5y}\, =\, \frac{1}{15};\quad \frac{1}{2x}\, +\, \frac{1}{3y}\, =\, \frac{1}{12}$$

A
$$x = 5, y = 3$$

B
$$x = 6, y = 6$$

C
$$x = 2, y = -2$$

D
$$x = 1, y = -5$$

Solution

On putting $$\dfrac1x=X$$ and $$\dfrac1y=Y$$ in given equations, we get

$$\dfrac X3+\dfrac Y5=\dfrac{1}{15}\Rightarrow 5X+3Y=1$$            ...(i)

$$\dfrac X2+\dfrac Y3=\dfrac{1}{12}\Rightarrow 6X+4Y=1$$            ...(ii)
On multiplying (i) by 6 and (ii) by 5, we get
$$30X+18Y=6$$
$$\underline {\underset {-}30X\underset {-}{+}20Y=\underset{-}5}$$
             $$-2Y=1$$
$$\therefore Y=-\dfrac12$$
On putting $$Y=-\dfrac12$$ in (i), we get

$$5X+3\times-\dfrac12=1\Rightarrow 5X=\dfrac52\Rightarrow X=\dfrac12$$

$$\therefore x=\dfrac1X=2,y=\dfrac1Y=-2$$
Question 3
1 / -0

A man starts his job with a certain monthly salary and a fixed increment every year. If his salary will be Rs. $$11000$$ after $$2$$ years and Rs. $$14000$$ after $$4$$ years of his service. What is his starting salary and what is the annual increment?

A
Starting salary is Rs. $$7500$$ and increment is Rs. $$2500$$

B
Starting salary is Rs.$$5000$$ and increment is Rs. $$1800$$

C
Starting salary is Rs. $$8000$$ and increment is Rs. $$1500$$

D
Starting salary is Rs. $$7000$$ and increment is Rs. $$1300$$

Solution

Let starting salary $$=\ x$$
Increment every year $$=\ y$$
$$ x\ + 2y\ =\ 11,000\quad$$ ....... eq(1)
$$ x\ +\ 4y\ =\ 14,000$$ ......eq(2)
Substract eq(1) from eq(2)
$$2y\ =\ 3,000\\ y\ =\ 1500$$
Substitute $$ y =\ 1500$$ in eq(1), we get $$x\ =\ 11,000\ -\ 3,000$$
$$ x\ =\ 8,000$$
starting salary$$\ =\ 8,000$$
Increment $$\ = \ 1500$$
Question 4
1 / -0

Solve graphically the simultaneous equations given below. Take the scale as $$1 \ \mathrm{cm} = 1$$ unit on both the axes.
$$ x - 2y - 4=0 $$
$$ 2x + y= 3 $$

A
$$ x= 1,\:y= -1 $$

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

C
$$ x= 3,\:y= -1 $$

D
$$ x= 7,\:y= -1 $$

Solution

Plotting $$x-2y-4=0$$
For $$x=0$$ we get $$y=-2$$
For $$y=0$$ we get $$x=4$$
Join the two points $$(0,-2)$$ and $$(4,0)$$ by drawing line.
Plotting $$2x+y=3$$
For $$x=0$$ we get $$y=3$$
For $$y=0$$ we get $$x=1.5$$
Join the two points $$(0,3)$$ and $$(1.5,0)$$ by drawing line.
the point of intersection of both lines is $$(2,-1)$$
Question 5
1 / -0

Solve the given pair of equations by substitution method:
$$x\, +\, 5y\, =\, 18$$
$$3x\, +\, 2y\, =\, 41$$

A
$$(13 , 1)$$

B
$$(1 , 3)$$

C
$$(12 , 17)$$

D
$$(9 , 16)$$

Solution

We have, $$x+5y =18\Rightarrow x = 18-5y$$ ..........(1)
and $$3x+2y=41$$.......(2)
Now substitute value x in terms of y from (1) to (2)
$$\Rightarrow 3(18-5y)+2y=41$$
$$\Rightarrow 54-15y+2y=41$$
$$\Rightarrow 15y-2y=54-41$$
$$\Rightarrow 13y=13\Rightarrow y=\dfrac{13}{13}=1$$
So $$x =18-5y=18-5(1)=13$$
Hence solution set is $$(x,y)=(13,1)$$
Question 6
1 / -0

Draw the graph of $$ 2x - y -1=0 $$ and $$ 2x + y= 9 $$ on the same graph. Use $$2\ \mathrm{ cm} = 1$$ unit on both axes.
Write down the co-ordinates of the point of intersection of the two lines.

A
$$ x=3.5,\:y=4 $$

B
$$ x=4.5,\:y=4 $$

C
$$ x=1.5,\:y=4 $$

D
$$ x=2.5,\:y=4 $$

Solution

Plotting $$2x-y-1=0$$ and $$2x+y=9$$ in the graph, we get
the point of intersection as $$(2.5,4)$$
Question 7
1 / -0

Solve the following simultaneous equations.
$$3x+4y=18; \, \, 4x+3y=17$$

A
$$x = 2$$, $$y = 3$$

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

C
$$x=4$$, $$y=3$$

D
$$x=-2$$, $$y=-3$$

Solution

The given equations are
$$3x+4y=18$$                              ...(i)
$$4x+3y=17$$                           ...(ii)

On multiplying (i) by 4 and (ii) by 3 and subtracting, we get
$$12x+16y=72$$
$$\underline {\underset {-}12x\underset {-}{+}9y=\underset{-}51}$$
           $$7y=21$$
$$\Rightarrow y=3$$

On putting $$y=3$$ in (i), we get
$$3x+12=18\Rightarrow 3x=6\Rightarrow x=2$$

$$\therefore x=2,y=3$$
Question 8
1 / -0

Solve the given pair of equations by substitution method:
$$x\, +\, y\, =\, 11$$
$$x\, -\, y\, =\, -3$$

A
$$(4 , 6)$$

B
$$(3 , 11)$$

C
$$(4 , 7)$$

D
$$(6 , 2)$$

Solution

We pick either of the equations and write one variable in terms of the other.

Let us consider the equation $$x-y = -3$$ and write it as

$$x = -3+y...........(1)$$

Substitute the value of $$x$$ in Equation $$x+y = 11$$. We get

$$-3+y+y = 11$$

i.e. $$-3+y+y = 11$$

i.e. $$-3+2y = 11$$

i.e. $$2y = 14$$

Therefore, $$y = 7$$

Substituting this value of $$y$$ in Equation (1), we get

$$x = -3+7=4$$

Hence, the solution is $$x = 4, y = 7$$.
Question 9
1 / -0

Solve the given pair of equations by substitution method:
$$4a\, -\, b\, =\, 10$$
$$2a\, +\, 3b\, =\, 12$$

A
$$a = 0, b = 5$$

B
$$a = 7, b = 6$$

C
$$a = 1, b = 7$$

D
$$a = 3, b = 2$$

Solution

We have, $$4a-b=10\Rightarrow b=4a-10$$.........(1)
and $$2a+3b=12$$........(2)
Now substitute value of b in terms of a in (2)
$$\Rightarrow 2a+3(4a-10)=12$$
$$\Rightarrow 2a+12a-30=12$$
$$\Rightarrow 14a=12+30=42\Rightarrow a=\dfrac{42}{14}=3$$
So $$b=4a-10=4(3)-10=12-10=2$$

Hence option 'D' is correct choice
Question 10
1 / -0

Solve the given pair of equations by substitution method:
$$x\, -\, 4y\, =\, -8$$
$$x\, -\, 2y\, =\, 0$$

A
$$(2 , -1)$$

B
$$(7, -6)$$

C
$$(8 , 4)$$

D
$$(-3 , 6)$$

Solution

We have, $$x-4y =-8\Rightarrow x = 4y-8$$ ..........(1)
and $$x-2y=0$$.......(2)
Now substitute value of x in terms of y from (1) to (2)
$$\Rightarrow 4y-8-2y=0$$
$$\Rightarrow 2y=8$$
$$\Rightarrow y=\dfrac{8}{2}=4$$
So $$x =4y-8=4(4)-8=8$$
Hence solution set is $$(x,y)=(8,4)$$

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Pair of Linear Equations in Two Variables Test - 25

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Pair of Linear Equations in Two Variables Test - 25

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Question 1

$$75, 14$$

$$90, 14$$

$$75, 18$$

$$85, 12$$

Solution

Question 2

$$x = 5, y = 3$$

$$x = 6, y = 6$$

$$x = 2, y = -2$$

$$x = 1, y = -5$$

Solution

Question 3

Starting salary is Rs. $$7500$$ and increment is Rs. $$2500$$

Starting salary is Rs.$$5000$$ and increment is Rs. $$1800$$

Starting salary is Rs. $$8000$$ and increment is Rs. $$1500$$

Starting salary is Rs. $$7000$$ and increment is Rs. $$1300$$

Solution

Question 4

$$ x= 1,\:y= -1 $$

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

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

$$ x= 7,\:y= -1 $$

Solution

Question 5

$$(13 , 1)$$

$$(1 , 3)$$

$$(12 , 17)$$

$$(9 , 16)$$

Solution

Question 6

$$ x=3.5,\:y=4 $$

$$ x=4.5,\:y=4 $$

$$ x=1.5,\:y=4 $$

$$ x=2.5,\:y=4 $$

Solution

Question 7

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

$$x = -2$$, $$y = 5$$

$$x=4$$, $$y=3$$

$$x=-2$$, $$y=-3$$

Solution

Question 8

$$(4 , 6)$$

$$(3 , 11)$$

$$(4 , 7)$$

$$(6 , 2)$$

Solution

Question 9

$$a = 0, b = 5$$

$$a = 7, b = 6$$

$$a = 1, b = 7$$

$$a = 3, b = 2$$

Solution

Question 10

$$(2 , -1)$$

$$(7, -6)$$

$$(8 , 4)$$

$$(-3 , 6)$$

Solution

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