Linear Equations in One Variable is an important chapter of Class 8 Mathematics. In this chapter, students learn how to find an unknown value using mathematical equations. The basic idea is simple: an equation has two equal sides, and whatever operation is performed on one side must also be performed on the other side. This complete guide explains linear equations in one variable formulas, important rules, solved questions, shortcuts, important notes, and exam tips in easy English.
Students can use this guide for school exams, homework, revision, and understanding the basic concepts of algebra. The article covers important formulas, transposition rules, equations with brackets, equations with fractions, equations having variables on both sides, and word problems.
Linear Equations in One Variable: Formulas, Important Questions and Notes
A linear equation becomes easier when we understand the meaning of equality. An equation can be compared with a balanced scale. If something is added, subtracted, multiplied, or divided on one side, the same operation must be performed on the other side.
The main goal is to find the value of the variable. Usually, the variable is represented by a letter such as x.
What Is a Linear Equation in One Variable?
A linear equation in one variable is an equation that contains one variable and the highest power of that variable is 1.
Here, x is the variable and its highest power is 1. Therefore, 3x + 5 = 20 is a linear equation in one variable.
Some examples of linear equations are:
- x + 8 = 15
- 5x = 40
- 4x - 7 = 21
- 7x + 3 = 4x + 18
- 2(x + 5) = 24
However, x² + 5 = 14 is not a linear equation because the highest power of x is 2.
Important Terms Used in Linear Equations
- Variable: An unknown quantity represented by a letter such as x, y, or z.
- Constant: A fixed number such as 5, 10, 20, or 100.
- Coefficient: The number multiplied by a variable. In 7x, 7 is the coefficient.
- LHS: The expression present on the left-hand side of the equal sign.
- RHS: The expression present on the right-hand side of the equal sign.
- Equation: A mathematical statement containing an equal sign.
All Important Formulas of Linear Equations in One Variable
The following formulas are useful for solving different forms of linear equations. Students should understand how each formula is obtained instead of simply memorising it.
1. Formula: x + a = b
x = b - a
Example:
x = 15 - 7
x = 8
2. Formula: x - a = b
x = b + a
Example:
x = 14 + 6
x = 20
3. Formula: ax = b
x = b/a
Example:
x = 35/5
x = 7
4. Formula: x/a = b
x = ab
Example:
x = 8 × 5
x = 40
5. Formula: ax + b = c
x = (c - b)/a
6. Formula: ax - b = c
x = (c + b)/a
7. Formula: ax + b = cx + d
(a - c)x = d - b
Therefore,
x = (d - b)/(a - c)
8. Formula: ax - b = cx + d
(a - c)x = b + d
Therefore,
x = (b + d)/(a - c)
9. Distributive Property
This rule is very important when an equation contains brackets.
10. Fraction Rule
If an equation contains fractions, multiply both sides by the required denominator to remove the fractions.
x = ab
Rules for Solving Linear Equations
The most important principle in solving an equation is maintaining equality. Both sides of an equation must always remain equal.
- Add the same number to both sides.
- Subtract the same number from both sides.
- Multiply both sides by the same non-zero number.
- Divide both sides by the same non-zero number.
Transposition Rule
Transposition is a shortcut method used while solving equations. When a term is moved from one side of the equation to the other side, its operation changes.
- + becomes −
- − becomes +
- × becomes ÷
- ÷ becomes ×
x = 20 - 9
x = 11
x = 20 + 9
x = 29
x = 28/4
x = 7
How to Solve a Linear Equation Step by Step
- Read the equation carefully.
- Simplify brackets if they are present.
- Collect variable terms on one side.
- Collect constant terms on the other side.
- Simplify both sides.
- Divide by the coefficient of the variable.
- Check the answer by substituting it into the original equation.
15 Important Questions with Solutions
Question 1: Solve x + 7 = 15.
x + 7 = 15
x = 15 - 7
x = 8
Answer: x = 8
Question 2: Solve x - 9 = 13.
x - 9 = 13
x = 13 + 9
x = 22
Answer: x = 22
Question 3: Solve 5x = 45.
5x = 45
x = 45/5
x = 9
Answer: x = 9
Question 4: Solve x/4 = 6.
x/4 = 6
x = 6 × 4
x = 24
Answer: x = 24
Question 5: Solve 3x + 5 = 20.
3x + 5 = 20
3x = 20 - 5
3x = 15
x = 15/3
x = 5
Answer: x = 5
Question 6: Solve 7x - 4 = 31.
7x - 4 = 31
7x = 31 + 4
7x = 35
x = 35/7
x = 5
Answer: x = 5
Question 7: Solve 4x + 3 = 2x + 15.
4x + 3 = 2x + 15
4x - 2x = 15 - 3
2x = 12
x = 12/2
x = 6
Answer: x = 6
Question 8: Solve 5x - 7 = 3x + 9.
5x - 7 = 3x + 9
5x - 3x = 9 + 7
2x = 16
x = 16/2
x = 8
Answer: x = 8
Question 9: Solve 2(x + 3) = 18.
2(x + 3) = 18
2x + 6 = 18
2x = 18 - 6
2x = 12
x = 12/2
x = 6
Answer: x = 6
Question 10: Solve 3(x - 2) = 21.
3(x - 2) = 21
3x - 6 = 21
3x = 21 + 6
3x = 27
x = 27/3
x = 9
Answer: x = 9
Question 11: Solve 2x/3 = 10.
2x/3 = 10
2x = 30
x = 30/2
x = 15
Answer: x = 15
Question 12: Solve (x + 5)/3 = 7.
(x + 5)/3 = 7
x + 5 = 21
x = 21 - 5
x = 16
Answer: x = 16
Question 13: Solve 4x + 7 = 3x + 18.
4x + 7 = 3x + 18
4x - 3x = 18 - 7
x = 11
Answer: x = 11
Question 14: The sum of a number and 12 is 30. Find the number.
Let the number be x.
According to the question:
x + 12 = 30
x = 30 - 12
x = 18
Answer: The number is 18.
Question 15: Three times a number decreased by 5 is 25. Find the number.
Let the number be x.
Three times the number = 3x.
According to the question:
3x - 5 = 25
3x = 25 + 5
3x = 30
x = 30/3
x = 10
Answer: The number is 10.
Important Notes for Exams
- Always identify the variable before solving.
- Remember that both sides of an equation must remain equal.
- Perform the same operation on both sides.
- Be careful when changing signs during transposition.
- Combine only like terms.
- Use the distributive property correctly when removing brackets.
- Pay special attention to negative signs.
- For fractions, remove denominators carefully.
- Always check the final answer in the original equation.
- For word problems, first assume the unknown quantity as x.
Common Mistakes Students Should Avoid
- Changing the sign incorrectly while transposing a term.
- Forgetting to multiply every term inside a bracket.
- Dividing only one side of an equation.
- Combining unlike terms.
- Ignoring a negative sign.
- Writing the final answer without checking it.
- Confusing the coefficient with the constant.
Quick Formula Revision Table
| Equation | Formula / Result |
|---|---|
| x + a = b | x = b - a |
| x - a = b | x = b + a |
| ax = b | x = b/a |
| x/a = b | x = ab |
| ax + b = c | x = (c - b)/a |
| ax - b = c | x = (c + b)/a |
| ax + b = cx + d | x = (d - b)/(a - c) |
| a(b + c) | ab + ac |
| a(b - c) | ab - ac |
How to Solve Word Problems Using Linear Equations
Linear equations are very useful for solving mathematical word problems. The most important step is to convert the information given in the question into an equation.
For example, suppose the question says: “The sum of a number and 8 is 25. Find the number.”
First, let the unknown number be x.
Now solve the equation:
x = 17
Therefore, the unknown number is 17.
Why Is This Chapter Important?
Linear Equations in One Variable provides the foundation for algebra. The concepts learned in this chapter are useful in higher classes when students study more advanced algebraic equations, graphs, and other mathematical topics.
Linear equations can also be used in practical situations to find an unknown price, age, distance, quantity, number, or measurement. Therefore, understanding the method of solving equations is more important than simply memorising formulas.
Frequently Asked Questions
What is a linear equation in one variable?
A linear equation in one variable is an equation containing one variable whose highest power is 1. For example, 2x + 7 = 15 is a linear equation in one variable.
What is the most important rule for solving an equation?
The most important rule is to perform the same mathematical operation on both sides of the equation.
What is transposition?
Transposition is a shortcut method in which a term is moved from one side of an equation to the other side by changing its operation or sign.
How can I check my answer?
Substitute the value obtained for the variable into the original equation. If the LHS and RHS have equal values, the solution is correct.
Is x² + 5 = 14 a linear equation?
No. It is not a linear equation because the highest power of x is 2.
How do I solve an equation containing brackets?
First expand the brackets using the distributive property, simplify the equation, collect variable and constant terms, and then find the value of the variable.
How do I solve an equation with variables on both sides?
Bring the variable terms to one side and the constant terms to the other side. Then simplify and divide by the coefficient of the variable.
Conclusion
Linear Equations in One Variable is a fundamental Class 8 Maths chapter. The main concepts include variables, constants, coefficients, equality, transposition, brackets, fractions, and word problems. Students should understand the important formulas and practise different types of equations to improve their problem-solving skills.
The most useful formulas include x + a = b, x - a = b, ax = b, x/a = b, ax + b = c, and ax + b = cx + d. Along with these formulas, students should remember that both sides of an equation must remain equal.
Regular practice of Class 8 Linear Equations in One Variable important questions can help students become more confident and accurate. Always solve the equation step by step and check the final value in the original equation.
Disclaimer
This article is provided for educational and revision purposes. The examples and explanations are designed to help Class 8 students understand the basic concepts of linear equations. Students should also follow their school textbook, teacher's instructions, and prescribed syllabus for examination preparation.