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What is Linear Equations in One Variable?


A linear equation in one variable is an equation which has a maximum of one variable of order 1. It is of the form ax + b = 0, where x is the variable.
This equation has only one solution. A few examples are:

  1. 3x = 1
  2. 22x - 1 =0
  3. 4x + 9 = -11

Solved Examples


Q1:  The solution of 2x-3=7 is:
(a) 5
(b) 7
(c) 12
(d) 11
Ans: 
(a)

2x-3=7
2x=7+3
=10
x=10/2
= 5

Q2: 3/4 part of a number is 5 more than its 2/3 parts. This statement in the form of an equation is
(a) 2/3 x – 3/4 x = 5
(b) 2/3 x – 5 = 3/4 x
(c) 3/4 x = 2/3 x + 5
(d) 3/4 x – 5 = –2/3 x
Ans:
(a)

Let's represent the unknown number as x. The statement "3/4 part of a number is 5 more than its 2/3 parts" can be translated into the equation:
¾ x = 2/3 x + 5
In this equation:

  • 3/4x represents 3/4 part of the number,
  • 2/3x represents 2/3 parts of the number, and
  • 5 is the additional amount.

Q3: The ratio of boys and girls in the class is 9:5, the number of boys is 12 more than the number of girls. What is the strength of the class?
(a) 40
(b) 41
(c) 42
(d) 45
Ans:
(c)

let the number of boys and girls in the class be 9x and 5x respectively.
According to the question 9x + 12 = 5x
4x = 12
X = 3
The total strength of class=no. of boys + no. of girls
= 9x + 5x
= 14x
= 14 x 3
= 42

Q4: Solve 3y + 6/5 = 27/5 - y
(a) 2/5
(b) 21/4
(c) 27/8
(d) 21/20
Ans: 
(d)

3y + 6/5 = 27/5 - y
3y + y = 27/5 - 6/5
4y = 21/5
y=21/20

Q5: Solve: 3b=5b – 8/5
(a) 4/5
(b) 6/5
(c) 3/5
(d) 9/5
Ans: 
(a)

3b = 5b – 8/5
2b = 8/5
b = 4/5

Q6: Simplify the following linear equation:
4(m - 4) = 5(2m - 5)
(a) 3/2
(b) 5/2
(c) 9/2
(d) 7/1
Ans:
(a)

4(m - 4) = 5(2m - 5)
4m - 16 = 10m - 25
6m = 9
m  = 3/2

Q7: Present ages of Akansha and Anubhav are in the ratio of 5/7. Four years from now the ratio of their ages will be 7/9. Find the present age of Akansha.
(a) 12
(b) 10
(c) 14
(d) 18
Ans:
(b)

Let the present age of Akansha and Anubhav be 5x and 7x years respectively.
After four years their ages will be 5x + 4, 7x + 4
The ratio of their ages after 4 years will be
5x + 4/7x + 4 = 7/9
By cross multiplication
9(5x + 4) = 7(7x + 4)
45x + 36 = 49x + 28
(49 - 45)x = 36 - 28
4x = 8
x = 2
Therefore, Akansha’s age is 5x i.e., 5 x 2 = 10 years.

Q8: When a number is added to itself, it becomes 24. What is the number?
(a) 2
(b) 4
(c) 12
(d) 21
Ans:
(c)

Let the number be x.
x + x = 24
2x = 24
x = 24/2
x = 12

Q9: The difference between the two numbers is 30. If the bigger number is x, then what is the smaller number?
(a) x – 30
(b) 30 – x
(c) 30x
(d) None of these
Ans: 
(a)

x – small number = 30
Small number = x – 30

Q10: If x is an even number, then the next even number is:
(a) x + 1
(b) x + 2
(c) x + 3
(d) x + 4
Ans:
(b)

If x = 2, then x + 2 = 2 + 2 = 4

Q11: The perimeter of a rectangle is 40 cm. If its width is 10 cm, then find the length.
(a) 10
(b) 20
(c) 30
(d) 40
Ans: 
(a)

Perimeter of a rectangle = 40 cm
Width = 10 cm
Let the length be x.
Perimeter of rectangle = 2(length + width)
40 = 2 (x + 10)
40/2 = x + 10
20 = x + 10
x = 20 – 10 = 10
Thus, the length is also 10 cm.
Hence, we can say, that the given rectangle is a square, with all its sides equal.

Q12: Find the value of x if 2x + 10 = 76.
(a) 33
(b) 7.6
(c) 66
(d) 32
Ans: 
(a)

2x + 10 = 76
2x = 76 – 10
2x = 66
x = 66/2
x = 33

Q13: Solve 2x + 9 = 4.
(a) X = 6
(b) X = -5/2
(c) X = -3/2
(d) X = -9/2
Ans:
(b)

2x + 9 = 4
2x = 4 – 9
2x = -5
x = -5/2

Q14: If a number is divided by 8 it gives 6 as the value. Find the number.
(a) 36
(b) 42
(c) 48
(d) 56
Ans:
(c)

Let X be the number
X/8 = 6
X = 8 x 6 = 48

Q15: What is the value of x if x + 9 = 12?
(a) 2
(b) 3
(c) 8
(d) 6
Ans: 
(b)

x + 9 = 12
X = 12 – 9
X = 3

Q16: The solution for 3m = 5m – (8/5) is:
(a) 8/5
(b) 4/5
(c) 5/4
(d) 4/3
Ans: 
(b)

3m = 5m – (8/5)
8/5 = 5m – 3m
2m = 8/5
m = 8/10 = 4/5

Q17: Three consecutive integers add up to 51. The integers are:
(a) 16,17,18
(b) 15,16,17
(c) 17,18,19
(d) 18,19,20
Ans:
(a)

Let the three consecutive integers be x, x + 1, x + 2
x + (x + 1) + (x + 2) = 51
3x + 3 = 51
3x = 51 – 3
x = 48/3 = 16
x + 1 = 16 + 1=17
x + 2 = x + 2 = 18

Q18: The difference between the two whole numbers is 66. The ratio of the two numbers is 2: 5. The two numbers are:
(a) 60 and 6
(b) 100 and 33
(c) 110 and 44
(d) 99 and 33
Ans:
(c)

Let the two numbers be 2x and 5x since they are in the ratio of 2:5.
The difference between 5x and 2x = 66
5x – 2x = 66
3x = 66
x = 22
Hence, 2x = 2(22) = 44 and 5x = 5(22) = 110.

Q19: The age of the father is three times the age of the son. If the age of the son is 15 years old, then the age of the father is:
(a) 50 years
(b) 55 years
(c) 40 years
(d) 45 years
Ans: 
(d)

Let age of the father is x
Given: x = 3 × (age of son) = 3 × (15) = 45 years

Q20: The perimeter of a rectangle is 20cm. If the length of the rectangle is 6cm, then its breadth will be:
(a) 4 cm
(b) 6 cm
(c) 10 cm
(d) 14 cm
Ans:
(a)

Perimeter of rectangle = 2(Length + Breadth)
20 = 2(6+x)
6 + x = 20/2
6 + x = 10
x = 10-6
x = 4 cm

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