Solving Linear Equations in One Variable | Complete Guide with Examples, Short Notes, MCQs & Worksheet

Introduction

The graphical method is particularly useful for developing a visual understanding of linear equations.

Common Mistakes in Solving Linear Equations

Students often make the following mistakes:

1. Changing the sign incorrectly

If:$$x-5=10$$

then adding 5 gives:$$x=15$$

not $x=5$.

2. Performing an operation on only one side

If you subtract 4 from the left side, you must also subtract 4 from the right side.

3. Making mistakes while removing brackets

Remember:$$a(b+c)=ab+ac$$

and$$a(b-c)=ab-ac$$

4. Forgetting to check the answer

Always substitute your answer into the original equation whenever possible.

Short Notes on Solving Linear Equations

• A linear equation in one variable has the variable with power 1.
• The objective is to find the value of the variable.
• Perform the same operation on both sides of an equation.
• Add or subtract constants to isolate the variable term.
• Divide or multiply to obtain the value of the variable.
• Remove brackets carefully using the distributive property.
• Simplify fractions when necessary.
• Collect like terms before solving.
• Variables can occur on either side of an equation.
• Always check the final answer by substitution.
• Graphically, the solution of two linear equations can be represented by their point of intersection.

Formula and Basic Rules

If:$$x+a=b$$

then:$$x=b-a$$

If:$$x-a=b$$

then:$$x=b+a$$

If:$$ax=b$$

then:$$x=\frac{b}{a},\quad a\ne0$$

If:$$\frac{x}{a}=b$$

then:$$x=ab$$

These basic forms can help students solve many linear equations quickly.

MCQs on Solving Linear Equations

MCQ 1

What is the solution of:$$x+5=12$$

A. 5
B. 6
C. 7
D. 8

Answer: C. 7

MCQ 2

What is the solution of:$$3x=21$$

A. 6
B. 7
C. 8
D. 9

Answer: B. 7

MCQ 3

Solve:$$x-8=4$$

A. 4
B. 8
C. 12
D. 14

Answer: C. 12

MCQ 4

Which of the following is a linear equation in one variable?

A. $x^2+3=7$
B. $2x+5=11$
C. $xy=10$
D. $x^3=8$

Answer: B. 2x+5=112x+5=11

MCQ 5

The solution of $5x=40$ is:

A. 5
B. 6
C. 7
D. 8

Answer: D. 8

MCQ 6

Solve:$$2x+4=14$$

A. 4
B. 5
C. 6
D. 7

Answer: B. 5

MCQ 7

If:$$\frac{x}{3}=9$$

then $x$ is:

A. 3
B. 18
C. 27
D. 30

Answer: C. 27

MCQ 8

What should be done to both sides of $x+6=15$ to find $x$?

A. Add 6
B. Subtract 6
C. Multiply by 6
D. Divide by 6

Answer: B. Subtract 6

MCQ 9

The graph of a linear equation in two variables is generally a:

A. Circle
B. Parabola
C. Straight line
D. Triangle

Answer: C. Straight line

MCQ 10

What is the first step in solving:$$4x+7=23$$

A. Divide by 4
B. Add 7
C. Subtract 7
D. Multiply by 4

Answer: C. Subtract 7

Worksheet / Assignment

Topic: Solving Linear Equations in One Variable

Part A: Solve the following equations

1. $x+9=17$
2. $x-6=14$
3. $4x=36$
4. $7x=49$
5. $2x+5=19$
6. $3x-4=20$
7. $5x+8=33$
8. $6x-7=29$
9. $2x+3=x+10$
10. $5x-4=2x+14$

Part B: Solve equations containing brackets

11. $3(x+2)=21$
12. $4(x-3)=20$
13. $2(x+5)+3=17$
14. $5(x-2)-4=16$
15. $3(x+4)-2=19$

Part C: Solve equations containing fractions

16. $\frac{x}{2}=8$
17. $\frac{x}{5}=7$
18. $\frac{x+2}{3}=6$
19. $\frac{x-4}{2}=9$
20. $\frac{2x+4}{3}=8$

Part D: Word Problems

21. A number increased by 8 is 25. Find the number.
22. Three times a number is 36. Find the number.
23. A number decreased by 7 is 15. Find the number.
24. Five times a number plus 4 is 29. Find the number.
25. The sum of a number and twice the same number is 24. Find the number.

Challenge Questions

26. $7x+5=3x+25$
27. $9x-8=4x+17$
28. $3(2x-1)=21$
29. $4(x+3)-5=19$
30. $2(3x+4)=4x+20$

Answers to Worksheet

1. $x=8$
2. $x=20$
3. $x=9$
4. $x=7$
5. $x=7$
6. $x=8$
7. $x=5$
8. $x=6$
9. $x=7$
10. $x=6$
11. $x=5$
12. $x=8$
13. $x=2$
14. $x=6$
15. $x=3$
16. $x=16$
17. $x=35$
18. $x=16$
19. $x=22$
20. $x=10$
21. $x=17$
22. $x=12$
23. $x=22$
24. $x=5$
25. $x=8$
26. $x=5$
27. $x=5$
28. $x=4$
29. $x=3$
30. $x=6$

Learning Support from Nisar Math Academy

Students looking for additional mathematics learning resources can visit Nisar Math Academy for recorded lectures, mathematics notes, assessments, lesson plans and educational blog posts.

The academy also provides selected free learning resources, including Class 9 Maths Notes for the first exercise of each chapter and free lectures covering selected questions. Students who want complete access to the full course can explore the membership option.

Website: www.nisarmathacademy.com

Conclusion

Solving linear equations is an essential algebraic skill. Once students understand the principle of keeping both sides of an equation balanced, even equations involving brackets, fractions and variables on both sides become much easier.

The best way to master this topic is to understand the method first and then solve a large number of practice questions. Students should also develop the habit of checking their answers by substituting the obtained value back into the original equation.

With regular practice and proper guidance from Sir Nisar and Nisar Math Academy, students can build a strong foundation in algebra and prepare themselves for more advanced mathematical concepts.

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