Class 8 Mathematics Chapter 2 Linear Equations in One Variable

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Introduction to Linear Equations in One Variable: A linear equation in one variable is an equation that involves only one variable and has the highest exponent of that variable as 1. These equations are of the form:ax+b=0ax + b = 0ax+b=0

where aaa and bbb are constants, and xxx is the variable.

In this blog post, we will discuss all the exercises from Chapter 2 of the Class 8 mathematics curriculum. After going through the exercises, we will also provide solutions to each problem to ensure a better understanding of the topic. Additionally, 40 extra questions will be provided at the end for further practice.

Exercise 2.1: Basic Linear Equations

Q1. Solve the following equation:3x−7=113x – 7 = 113x−7=11

Solution:

Step 1: Add 7 to both sides.3x=11+7=183x = 11 + 7 = 183x=11+7=18

Step 2: Divide both sides by 3.x=183=6x = \frac{18}{3} = 6x=318​=6

Answer: x=6x = 6x=6

Q2. Solve:5x+8=185x + 8 = 185x+8=18

Solution:

Step 1: Subtract 8 from both sides.5x=18−8=105x = 18 – 8 = 105x=18−8=10

Step 2: Divide both sides by 5.x=105=2x = \frac{10}{5} = 2x=510​=2

Answer: x=2x = 2x=2

Q3. Solve the equation:4(x−2)=124(x – 2) = 124(x−2)=12

Solution:

Step 1: Distribute the 4 on the left side.4x−8=124x – 8 = 124x−8=12

Step 2: Add 8 to both sides.4x=12+8=204x = 12 + 8 = 204x=12+8=20

Step 3: Divide both sides by 4.x=204=5x = \frac{20}{4} = 5x=420​=5

Answer: x=5x = 5x=5

Exercise 2.2: Word Problems Based on Linear Equations

Q1. A number is 7 more than three times another number. If the sum of these numbers is 32, find the numbers.

Solution:

Let the first number be xxx and the second number be yyy.

According to the problem, we have the equation:x=3y+7x = 3y + 7x=3y+7

Also, the sum of the numbers is 32:x+y=32x + y = 32x+y=32

Substitute x=3y+7x = 3y + 7x=3y+7 into the second equation:(3y+7)+y=32(3y + 7) + y = 32(3y+7)+y=32

Simplify:4y+7=324y + 7 = 324y+7=32

Step 1: Subtract 7 from both sides.4y=254y = 254y=25

Step 2: Divide both sides by 4.y=254=6.25y = \frac{25}{4} = 6.25y=425​=6.25

Now, substitute y=6.25y = 6.25y=6.25 back into x=3y+7x = 3y + 7x=3y+7:x=3(6.25)+7=18.75+7=25.75x = 3(6.25) + 7 = 18.75 + 7 = 25.75x=3(6.25)+7=18.75+7=25.75

Answer: The numbers are 25.7525.7525.75 and 6.256.256.25.

Q2. A rectangle’s length is 3 more than twice its width. If the perimeter of the rectangle is 46 meters, find the length and width.

Solution:

Let the width of the rectangle be www meters. Then, the length is 2w+32w + 32w+3 meters.

The perimeter of a rectangle is given by the formula:Perimeter=2(length+width)\text{Perimeter} = 2(\text{length} + \text{width})Perimeter=2(length+width)

Substitute the values:46=2((2w+3)+w)46 = 2((2w + 3) + w)46=2((2w+3)+w)

Simplify:46=2(3w+3)46 = 2(3w + 3)46=2(3w+3)

Step 1: Divide both sides by 2.23=3w+323 = 3w + 323=3w+3

Step 2: Subtract 3 from both sides.20=3w20 = 3w20=3w

Step 3: Divide both sides by 3.w=203=6.67w = \frac{20}{3} = 6.67w=320​=6.67

Now substitute w=6.67w = 6.67w=6.67 into the length equation:Length=2(6.67)+3=13.34+3=16.34\text{Length} = 2(6.67) + 3 = 13.34 + 3 = 16.34Length=2(6.67)+3=13.34+3=16.34

Answer: The length is 16.34 meters, and the width is 6.67 meters.

Exercise 2.3: Linear Equations with Fractions

Q1. Solve the equation:2×3+5=9\frac{2x}{3} + 5 = 932x​+5=9

Solution:

Step 1: Subtract 5 from both sides.2×3=9−5=4\frac{2x}{3} = 9 – 5 = 432x​=9−5=4

Step 2: Multiply both sides by 3 to eliminate the fraction.2x=122x = 122x=12

Step 3: Divide both sides by 2.x=122=6x = \frac{12}{2} = 6x=212​=6

Answer: x=6x = 6x=6

Q2. Solve the equation:x4−3=5\frac{x}{4} – 3 = 54x​−3=5

Solution:

Step 1: Add 3 to both sides.x4=5+3=8\frac{x}{4} = 5 + 3 = 84x​=5+3=8

Step 2: Multiply both sides by 4 to eliminate the fraction.x=8×4=32x = 8 \times 4 = 32x=8×4=32

Answer: x=32x = 32x=32

Additional Practice Questions (40 Questions)

1. Solve: 2x+5=152x + 5 = 152x+5=15
Solution:
Step 1: Subtract 5 from both sides:2x=15−5=102x = 15 – 5 = 102x=15−5=10

Step 2: Divide both sides by 2:x=102=5x = \frac{10}{2} = 5x=210​=5

Answer: x=5x = 5x=5

2. Solve: 4x−9=234x – 9 = 234x−9=23
Solution:
Step 1: Add 9 to both sides:4x=23+9=324x = 23 + 9 = 324x=23+9=32

Step 2: Divide both sides by 4:x=324=8x = \frac{32}{4} = 8x=432​=8

Answer: x=8x = 8x=8

3. Solve: 3×4=9\frac{3x}{4} = 943x​=9
Solution:
Step 1: Multiply both sides by 4:3x=9×4=363x = 9 \times 4 = 363x=9×4=36

Step 2: Divide both sides by 3:x=363=12x = \frac{36}{3} = 12x=336​=12

Answer: x=12x = 12x=12

4. Solve: 5(x+2)=255(x + 2) = 255(x+2)=25
Solution:
Step 1: Distribute the 5 on the left side:5x+10=255x + 10 = 255x+10=25

Step 2: Subtract 10 from both sides:5x=25−10=155x = 25 – 10 = 155x=25−10=15

Step 3: Divide both sides by 5:x=155=3x = \frac{15}{5} = 3x=515​=3

Answer: x=3x = 3x=3

5. Solve: 6x−7=296x – 7 = 296x−7=29
Solution:
Step 1: Add 7 to both sides:6x=29+7=366x = 29 + 7 = 366x=29+7=36

Step 2: Divide both sides by 6:x=366=6x = \frac{36}{6} = 6x=636​=6

Answer: x=6x = 6x=6

6. Solve: 2(x−3)=82(x – 3) = 82(x−3)=8
Solution:
Step 1: Distribute the 2 on the left side:2x−6=82x – 6 = 82x−6=8

Step 2: Add 6 to both sides:2x=8+6=142x = 8 + 6 = 142x=8+6=14

Step 3: Divide both sides by 2:x=142=7x = \frac{14}{2} = 7x=214​=7

Answer: x=7x = 7x=7

7. Solve: x+4=7x−10x + 4 = 7x – 10x+4=7x−10
Solution:
Step 1: Subtract xxx from both sides:4=6x−104 = 6x – 104=6x−10

Step 2: Add 10 to both sides:14=6×14 = 6×14=6x

Step 3: Divide both sides by 6:x=146=73x = \frac{14}{6} = \frac{7}{3}x=614​=37​

Answer: x=73x = \frac{7}{3}x=37​

8. Solve: 3x−5=10x+123x – 5 = 10x + 123x−5=10x+12
Solution:
Step 1: Subtract 3x3x3x from both sides:−5=7x+12-5 = 7x + 12−5=7x+12

Step 2: Subtract 12 from both sides:−17=7x-17 = 7x−17=7x

Step 3: Divide both sides by 7:x=−177x = \frac{-17}{7}x=7−17​

Answer: x=−177x = \frac{-17}{7}x=7−17​

9. Solve: 2x+3=4x−12x + 3 = 4x – 12x+3=4x−1
Solution:
Step 1: Subtract 2x2x2x from both sides:3=2x−13 = 2x – 13=2x−1

Step 2: Add 1 to both sides:4=2×4 = 2×4=2x

Step 3: Divide both sides by 2:x=42=2x = \frac{4}{2} = 2x=24​=2

Answer: x=2x = 2x=2

10. Solve: x−5=2x+4x – 5 = 2x + 4x−5=2x+4
Solution:
Step 1: Subtract xxx from both sides:−5=x+4-5 = x + 4−5=x+4

Step 2: Subtract 4 from both sides:−9=x-9 = x−9=x

Answer: x=−9x = -9x=−9

11. Solve: 7x+9=3x+217x + 9 = 3x + 217x+9=3x+21
Solution:
Step 1: Subtract 3x3x3x from both sides:4x+9=214x + 9 = 214x+9=21

Step 2: Subtract 9 from both sides:4x=124x = 124x=12

Step 3: Divide both sides by 4:x=124=3x = \frac{12}{4} = 3x=412​=3

Answer: x=3x = 3x=3

12. Solve: x+5=2x+9x + 5 = 2x + 9x+5=2x+9
Solution:
Step 1: Subtract xxx from both sides:5=x+95 = x + 95=x+9

Step 2: Subtract 9 from both sides:−4=x-4 = x−4=x

Answer: x=−4x = -4x=−4

13. Solve: 2×5=6\frac{2x}{5} = 652x​=6
Solution:
Step 1: Multiply both sides by 5:2x=6×5=302x = 6 \times 5 = 302x=6×5=30

Step 2: Divide both sides by 2:x=302=15x = \frac{30}{2} = 15x=230​=15

Answer: x=15x = 15x=15

14. Solve: 4x+5=104x + 5 = 104x+5=10
Solution:
Step 1: Subtract 5 from both sides:4x=10−5=54x = 10 – 5 = 54x=10−5=5

Step 2: Divide both sides by 4:x=54x = \frac{5}{4}x=45​

Answer: x=54x = \frac{5}{4}x=45​

15. Solve: 2x+1=3x+52x + 1 = 3x + 52x+1=3x+5
Solution:
Step 1: Subtract 2x2x2x from both sides:1=x+51 = x + 51=x+5

Step 2: Subtract 5 from both sides:−4=x-4 = x−4=x

Answer: x=−4x = -4x=−4

16. Solve: x+7=3x−2x + 7 = 3x – 2x+7=3x−2
Solution:
Step 1: Subtract xxx from both sides:7=2x−27 = 2x – 27=2x−2

Step 2: Add 2 to both sides:9=2×9 = 2×9=2x

Step 3: Divide both sides by 2:x=92x = \frac{9}{2}x=29​

Answer: x=92x = \frac{9}{2}x=29​

17. Solve: 5x+2=3x+185x + 2 = 3x + 185x+2=3x+18
Solution:
Step 1: Subtract 3x3x3x from both sides:2x+2=182x + 2 = 182x+2=18

Step 2: Subtract 2 from both sides:2x=162x = 162x=16

Step 3: Divide both sides by 2:x=162=8x = \frac{16}{2} = 8x=216​=8

Answer: x=8x = 8x=8

18. Solve: 8x−3=5x+98x – 3 = 5x + 98x−3=5x+9
Solution:
Step 1: Subtract 5x5x5x from both sides:3x−3=93x – 3 = 93x−3=9

Step 2: Add 3 to both sides:3x=123x = 123x=12

Step 3: Divide both sides by 3:x=123=4x = \frac{12}{3} = 4x=312​=4

Answer: x=4x = 4x=4

19. Solve: 6x−2=4x+86x – 2 = 4x + 86x−2=4x+8
Solution:
Step 1: Subtract 4x4x4x from both sides:2x−2=82x – 2 = 82x−2=8

Step 2: Add 2 to both sides:2x=102x = 102x=10

Step 3: Divide both sides by 2:x=102=5x = \frac{10}{2} = 5x=210​=5

Answer: x=5x = 5x=5

20. Solve: x+3=2x+7x + 3 = 2x + 7x+3=2x+7
Solution:
Step 1: Subtract xxx from both sides:3=x+73 = x + 73=x+7

Step 2: Subtract 7 from both sides:−4=x-4 = x−4=x

Answer: x=−4x = -4x=−4

21. Solve: 3x−4=5x+63x – 4 = 5x + 63x−4=5x+6
Solution:
Step 1: Subtract 3x3x3x from both sides:−4=2x+6-4 = 2x + 6−4=2x+6

Step 2: Subtract 6 from both sides:−10=2x-10 = 2x−10=2x

Step 3: Divide both sides by 2:x=−102=−5x = \frac{-10}{2} = -5x=2−10​=−5

Answer: x=−5x = -5x=−5

22. Solve: x−7=2x+5x – 7 = 2x + 5x−7=2x+5
Solution:
Step 1: Subtract xxx from both sides:−7=x+5-7 = x + 5−7=x+5

Step 2: Subtract 5 from both sides:−12=x-12 = x−12=x

Answer: x=−12x = -12x=−12

23. Solve: 3×2=6\frac{3x}{2} = 623x​=6
Solution:
Step 1: Multiply both sides by 2:3x=123x = 123x=12

Step 2: Divide both sides by 3:x=123=4x = \frac{12}{3} = 4x=312​=4

Answer: x=4x = 4x=4

24. Solve: 5x−3=2x+155x – 3 = 2x + 155x−3=2x+15
Solution:
Step 1: Subtract 2x2x2x from both sides:3x−3=153x – 3 = 153x−3=15

Step 2: Add 3 to both sides:3x=183x = 183x=18

Step 3: Divide both sides by 3:x=183=6x = \frac{18}{3} = 6x=318​=6

Answer: x=6x = 6x=6

25. Solve: 6x−5=2x+136x – 5 = 2x + 136x−5=2x+13
Solution:
Step 1: Subtract 2x2x2x from both sides:4x−5=134x – 5 = 134x−5=13

Step 2: Add 5 to both sides:4x=184x = 184x=18

Step 3: Divide both sides by 4:x=184=4.5x = \frac{18}{4} = 4.5x=418​=4.5

Answer: x=4.5x = 4.5x=4.5

26. Solve: x6+7=9\frac{x}{6} + 7 = 96x​+7=9
Solution:
Step 1: Subtract 7 from both sides:x6=9−7=2\frac{x}{6} = 9 – 7 = 26x​=9−7=2

Step 2: Multiply both sides by 6:x=2×6=12x = 2 \times 6 = 12x=2×6=12

Answer: x=12x = 12x=12

27. Solve: 3x+4=103x + 4 = 103x+4=10
Solution:
Step 1: Subtract 4 from both sides:3x=10−4=63x = 10 – 4 = 63x=10−4=6

Step 2: Divide both sides by 3:x=63=2x = \frac{6}{3} = 2x=36​=2

Answer: x=2x = 2x=2

28. Solve: 2(x−4)=102(x – 4) = 102(x−4)=10
Solution:
Step 1: Distribute the 2 on the left side:2x−8=102x – 8 = 102x−8=10

Step 2: Add 8 to both sides:2x=10+8=182x = 10 + 8 = 182x=10+8=18

Step 3: Divide both sides by 2:x=182=9x = \frac{18}{2} = 9x=218​=9

Answer: x=9x = 9x=9

29. Solve: 7x+9=5x+217x + 9 = 5x + 217x+9=5x+21
Solution:
Step 1: Subtract 5x5x5x from both sides:2x+9=212x + 9 = 212x+9=21

Step 2: Subtract 9 from both sides:2x=122x = 122x=12

Step 3: Divide both sides by 2:x=122=6x = \frac{12}{2} = 6x=212​=6

Answer: x=6x = 6x=6

30. Solve: 8x−3=2x+158x – 3 = 2x + 158x−3=2x+15
Solution:
Step 1: Subtract 2x2x2x from both sides:6x−3=156x – 3 = 156x−3=15

Step 2: Add 3 to both sides:6x=186x = 186x=18

Step 3: Divide both sides by 6:x=186=3x = \frac{18}{6} = 3x=618​=3

Answer: x=3x = 3x=3

31. Solve: x3=7\frac{x}{3} = 73x​=7
Solution:
Step 1: Multiply both sides by 3:x=7×3=21x = 7 \times 3 = 21x=7×3=21

Answer: x=21x = 21x=21

32. Solve: 5x−9=3x+175x – 9 = 3x + 175x−9=3x+17
Solution:
Step 1: Subtract 3x3x3x from both sides:2x−9=172x – 9 = 172x−9=17

Step 2: Add 9 to both sides:2x=262x = 262x=26

Step 3: Divide both sides by 2:x=262=13x = \frac{26}{2} = 13x=226​=13

Answer: x=13x = 13x=13

33. Solve: 3x+7=103x + 7 = 103x+7=10
Solution:
Step 1: Subtract 7 from both sides:3x=10−7=33x = 10 – 7 = 33x=10−7=3

Step 2: Divide both sides by 3:x=33=1x = \frac{3}{3} = 1x=33​=1

Answer: x=1x = 1x=1

34. Solve: 7x+5=4x+157x + 5 = 4x + 157x+5=4x+15
Solution:
Step 1: Subtract 4x4x4x from both sides:3x+5=153x + 5 = 153x+5=15

Step 2: Subtract 5 from both sides:3x=103x = 103x=10

Step 3: Divide both sides by 3:x=103x = \frac{10}{3}x=310​

Answer: x=103x = \frac{10}{3}x=310​

35. Solve: 3(x+4)=153(x + 4) = 153(x+4)=15
Solution:
Step 1: Distribute the 3 on the left side:3x+12=153x + 12 = 153x+12=15

Step 2: Subtract 12 from both sides:3x=33x = 33x=3

Step 3: Divide both sides by 3:x=33=1x = \frac{3}{3} = 1x=33​=1

Answer: x=1x = 1x=1

36. Solve: 2x−5=x+72x – 5 = x + 72x−5=x+7
Solution:
Step 1: Subtract xxx from both sides:x−5=7x – 5 = 7x−5=7

Step 2: Add 5 to both sides:x=12x = 12x=12

Answer: x=12x = 12x=12

37. Solve: x8=4\frac{x}{8} = 48x​=4
Solution:
Step 1: Multiply both sides by 8:x=4×8=32x = 4 \times 8 = 32x=4×8=32

Answer: x=32x = 32x=32

38. Solve: 5x+3=2x+125x + 3 = 2x + 125x+3=2x+12
Solution:
Step 1: Subtract 2x2x2x from both sides:3x+3=123x + 3 = 123x+3=12

Step 2: Subtract 3 from both sides:3x=93x = 93x=9

Step 3: Divide both sides by 3:x=93=3x = \frac{9}{3} = 3x=39​=3

Answer: x=3x = 3x=3

39. Solve: 4x−5=3x+94x – 5 = 3x + 94x−5=3x+9
Solution:
Step 1: Subtract 3x3x3x from both sides:x−5=9x – 5 = 9x−5=9

Step 2: Add 5 to both sides:x=14x = 14x=14

Answer: x=14x = 14x=14

40. Solve: 2x−3=3x+52x – 3 = 3x + 52x−3=3x+5
Solution:
Step 1: Subtract 2x2x2x from both sides:−3=x+5-3 = x + 5−3=x+5

Step 2: Subtract 5 from both sides:−8=x-8 = x−8=x

Answer: x=−8x = -8x=−8

Summary of Answers:

  1. x=5x = 5x=5
  2. x=8x = 8x=8
  3. x=12x = 12x=12
  4. x=3x = 3x=3
  5. x=6x = 6x=6
  6. x=7x = 7x=7
  7. x=73x = \frac{7}{3}x=37​
  8. x=−177x = \frac{-17}{7}x=7−17​
  9. x=2x = 2x=2
  10. x=−9x = -9x=−9
  11. x=3x = 3x=3
  12. x=−4x = -4x=−4
  13. x=15x = 15x=15
  14. x=54x = \frac{5}{4}x=45​
  15. x=−4x = -4x=−4
  16. x=92x = \frac{9}{2}x=29​
  17. x=8x = 8x=8
  18. x=4x = 4x=4
  19. x=5x = 5x=5
  20. x=−4x = -4x=−4
  21. x=−5x = -5x=−5
  22. x=−12x = -12x=−12
  23. x=4x = 4x=4
  24. x=6x = 6x=6
  25. x=4.5x = 4.5x=4.5
  26. x=12x = 12x=12
  27. x=2x = 2x=2
  28. x=9x = 9x=9
  29. x=6x = 6x=6
  30. x=3x = 3x=3
  31. x=21x = 21x=21
  32. x=13x = 13x=13
  33. x=1x = 1x=1
  34. x=103x = \frac{10}{3}x=310​
  35. x=1x = 1x=1
  36. x=12x = 12x=12
  37. x=32x = 32x=32
  38. x=3x = 3x=3
  39. x=14x = 14x=14
  40. x=−8x = -8x=−8

These solutions should help reinforce your understanding of solving linear equations! Feel free to reach out if you need further clarification on any of the steps.

Class 8 Maths Chapter 1: Rational Numbers Worksheet

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