Pair of Linear Equations in Two Variable Test

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Pair of Linear Equations in Two Variable Test - 3

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Question 1
1 / -0

Which of the following equations is not equivalent to 5x + 4y = 9?

A
15x + 12y = 27

B
6x + 7y = 8

C
25x + 20y = 45

D
30x + 24y = 54

Solution

The equation 6x + 7y = 8 is not equivalent to the given equation because .

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Question 2
1 / -0

Which of the following pairs of equations represents parallel lines?

A
x + y = 13, x - y = 1

B
2x + 3y = 5, 4x + 6y = 12

C
x + 3y = 7, 3x + y = 2

D
x + y = 4, x - y = 0

Solution

The two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 will be parallel, if .
, and ...for option (ii)

Hence, 2x + 3y = 5 and 4x + 6y = 12 are parallel lines.

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Question 3
1 / -0

Under which of the following circumstances does a pair of linear equations have no common solution?

A
When the lines representing the equations intersect at one point

B
When the lines representing the equations are coincident

C
When the lines representing the equations do not intersect

D
Both (2) and (3)

Solution

These two lines N and M do not intersect each other.
There is no common solution.

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Question 4
1 / -0

If 13y = 36 - 2x and 2y = 69 - 13x, find the value of 2x + 3y.

A
3

B
5

C
9

D
16

Solution

13y = 36 - 2x
Or 13y + 2x = 36 …… (1)
2y = 69 -13x
2y + 13x = 69 ….. (2)
Multiply equation (2) with 13 and equation (1) with 2.
26y + 4x = 72 …. (3)
26y + 169x = 897 … (4)
Subtract equation (4) from (3).
169x - 4x = 897 - 72
165x = 825
x =
x = 5
Put x = 5 in equation (1).
13y + 2 5 = 36
13y + 10 = 36
13y = 36 - 10 (Subtract 10 from both sides)
13y = 26
y = (Divide both sides by 13)
y = 2
Now, 2x + 3y = 2 5 + 3 2
= 10 + 6 = 16

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Question 5
1 / -0

What will be the point of intersection of x + y = 3 and 2x - 3y = 16?

A
(9, -13)

B
(6, 4)

C
(5, -2)

D
None of these

Solution

On solving the two equations, we get x = 5 and y = -2.

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Question 6
1 / -0

If a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 represent a pair of linear equations, which of the following expressions represents the value of x?

A

B

C

D

Solution

[a₁x + b₁y + c₁ = 0 ….. (i) ] b₂
[a₂x + b₂y + c₂ = 0 ..... (ii) ] b₁
Subtracting (ii) from (i), we get

x(a₁b₂ – a₂b₁) = b₁c₂ – b₂c₁
x =

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Question 7
1 / -0

Which of the following values of x and y represent the solution of the pair of equations 3x + 4y = 7 and x + 5y = 6?

A
x = 1, y = -1

B
x = 1, y = 1

C
x = , y = -1

D
x = -49, y = 11

Solution

3x + 4y = 7 ..... I
x + 5y = 6 …. II] 3
Now, subtracting II from I, we get
3x + 4y = 7
3x + 15y = 18
- - -

-11y = -11
y = 1

Now, putting the value of y in I, we get
3x + 4(1) = 7
3x = 3 x = 1
x = 1 and y = 1 represent the solution of the pair of equations 3x + 4y = 7 and x + 5y = 6.

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Question 8
1 / -0

Which of the following options would you use to eliminate x from the given system of equations?

x + 6y = 13 - - - - - (i)
3x + 4y = 11 - - - --(ii)

A
Multiplying equation (ii) by 3 and then subtracting (i) from (ii)

B
Multiplying equation (i) by 3 and then subtracting (ii) from (i)

C
Multiplying (i) by 3 and then adding (ii) to it

D
Multiplying (ii) by –1 and (i) by 3 and then subtracting (ii) from (i)

Solution

Multiplying equation (i) by 3 and then subtracting (ii) from (i) gives y as 2 and x as 1.

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Question 9
1 / -0

Which of the following values of x and y represent the solution of the pair of equations 4x + 4y = 8 and 3x + 6y = 9?

A
x = -1, y = 2

B
x = 1, y = 1

C
x = 2, y = -2

D
x = 4, y = 6

Solution

4x + 4y = 8
Or, x + y = 2 ...(i)
And 3x + 6y = 9
Or, x + 2y = 3 ...(ii)
Subtracting eq (i) from (ii), we get
y = 1
Substituting value of y in eq (i), we get
x = 1
Hence, x = 1 and y = 1 is the required answer.

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Question 10
1 / -0

The sum of the ages of two persons is 52 years and the difference between their ages is 26 years. What are their ages?

A
13 years and 39 years

B
14 years and 38 years

C
15 years and 37 years

D
16 years and 38 years

Solution

Let the ages of the two persons be x and y.
x + y = 52
x - y = 26
Adding these equations,
x + y + x - y = 52 + 26
2x = 78
x = 39
y = 52 - x
= 52 - 39
= 13
So, the ages of the persons are 13 and 39 years.

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Question 11
1 / -0

Half the difference of two numbers is 6 and the average of the numbers is 16. Find the numbers.

A
22, 10

B
18, 14

C
16, 12

D
14, 8

Solution

Let the two numbers be x and y.
(x - y) = 6
x - y = 12 ... (1)
= 16
Or x + y = 32 ... (2)
Add (1) and (2).
x - y = 12
x + y = 32
2x = 44
x = 22
Put x = 22 in equation (1).
22 - y = 12
y = 10

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Question 12
1 / -0

The sum of two numbers is 1. If 3 times the second number is added to the first number, the result equals 9. What are the two numbers?

A
-3, 4

B
-4, 5

C
-3, 2

D
3, -2

Solution

Let the two numbers be x and y.
x + y = 1 ... (1)
x + 3y = 9 ... (2)
Subtracting 1 from 2,
x + 3y = 9
-x - y = -1
2y = 8
y = 4
Putting y = 4 in equation 1,
x + 4 = 1
x = -3

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Question 13
1 / -0

The sum of a two-digit number and the number obtained by reversing its digits is 99. If the digits differ by 3, what is the product of the digits?

A
18

B
28

C
15

D
20

Solution

Let the digit at ones place be x and the digit at tens place be y.
Number formed = 10y + x
Now, number formed by reversing the digits = 10x + y
Sum = 10y + x + 10x + y
Or 11x + 11y = 99
Or x + y = 9 … (i)
Now,
x - y = 3 … (ii)
Adding equations (i) and (ii), we get
2x = 12
x = 6
y = 3
Number formed = 36
Product of digits = 18
If y - x = 3, then y = 6 and x = 3
So, the product is 18 in both cases.

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Question 14
1 / -0

One of the acute angles of a right-angled triangle measures x and the measure of the other acute angle is y (in degrees). If one of them measures 6° more than thrice the other angle, then which of the following are the correct equations?

A

B

C

D

Solution

x + y + 90° = 180° (Angle-sum property of triangle)
x + y = 90° ... (i)
Also,
y = 6° + 3x
Or, y - 3x = 6° … (ii)
Hence, (3) is the correct option.

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Question 15
1 / -0

Which of the following pairs of equations represents the reduced form of the equations and ?

A
3a + 2b = 12 and 2a + 3b = 13, where a = and b =

B
2a + 3b = 12 and 3a + 2b = 13, where a = and b =

C
2a + 3b = 13 and 3a + 2b = 12, where a = and b =

D
None of these

Solution

^.... I
…. II
Put = a and = b.
So,
I = 2
3a + 2b = 12
II
2a + 3b = 13

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