Linear Equations in One Variable (Class 8)
Linear Equations in One Variable is one of the most important chapters in Class 8 Mathematics. It introduces students to algebraic thinking and teaches them how to solve equations containing a single variable. This chapter lays a strong foundation for advanced mathematical concepts by developing logical reasoning and problem-solving skills through clear explanations, solved examples, and real-life applications.
Table of Contents
- Introduction
- What is a Variable?
- What is an Equation?
- Linear Equation in One Variable
- Steps to Solve Linear Equations
- Properties of Equality
- Solved Examples
- Word Problems
- Key Points
- Abbreviations
- HOTS Questions
- NCERT Exercise Questions & Answers
- One Shot Revision
- Tricky Questions for Viewers
Introduction
English
Mathematics is not only about numbers; it is also about finding unknown values and solving real-life problems. One of the most important concepts in algebra is the Linear Equation in One Variable. This chapter introduces students to equations that contain a single variable and teaches systematic methods to find its value.
A linear equation in one variable is an equation in which the highest power of the variable is 1. These equations are widely used in everyday situations such as calculating expenses, determining ages, finding distances, and solving practical problems involving unknown quantities.
In this chapter, students will learn the meaning of variables and equations, understand the concept of balance in an equation, and apply different operations to solve linear equations accurately. Through examples, word problems, and practice exercises, learners will develop logical thinking and problem-solving skills that form the foundation for advanced algebraic concepts in higher classes.
By the end of this chapter, students will be able to confidently solve linear equations, verify their answers, and apply algebraic methods to real-world situations.
हिन्दी
गणित केवल संख्याओं का अध्ययन नहीं है, बल्कि अज्ञात मानों को खोजने और वास्तविक जीवन की समस्याओं को हल करने का एक प्रभावी माध्यम भी है। Linear Equations in One Variable बीजगणित (Algebra) का एक महत्वपूर्ण अध्याय है, जो विद्यार्थियों को एक चर (Variable) वाले समीकरणों को समझने और हल करने की विधि सिखाता है।
इस अध्याय में हम Variable, Equation तथा Linear Equation की अवधारणाओं को समझेंगे और विभिन्न उदाहरणों एवं प्रश्नों के माध्यम से उनका समाधान करना सीखेंगे। यह अध्याय आगे आने वाले Algebra, Coordinate Geometry और Higher Mathematics के लिए मजबूत आधार तैयार करता है।
नियमित अभ्यास और सही समझ के माध्यम से विद्यार्थी न केवल समीकरणों को हल करना सीखेंगे, बल्कि अपनी तार्किक सोच (Logical Thinking) और समस्या समाधान क्षमता (Problem-Solving Skills) को भी विकसित करेंगे।
Examples
- x + 5 = 12
- 2x – 3 = 7
- 5y + 4 = 19
These are linear equations because the variable has power 1.
What is a Variable?
English
A variable is a symbol that represents an unknown value.
हिन्दी
Variable एक ऐसा चिन्ह होता है जो किसी अज्ञात संख्या को दर्शाता है।
Examples
- x
- y
- z
- a
Example:
x + 3 = 10
Here x is the variable.
What is an Equation?
English
A mathematical statement showing equality between two expressions is called an equation.
हिन्दी
दो व्यंजकों के बीच समानता दर्शाने वाले गणितीय कथन को Equation कहते हैं।
Example:
x + 4 = 9
Both sides are equal.
Linear Equation in One Variable
Definition
An equation containing only one variable with power 1 is called a Linear Equation in One Variable.
General Form
ax + b = c
Where:
- a ≠ 0
- x is variable
- b and c are constants
Examples
✔ 2x + 5 = 15
✔ x/3 + 5 = 11
Properties of Equality
Addition Property
If
x = y
then
x + a = y + a
Subtraction Property
If
x = y
then
x – a = y – a
Multiplication Property
If
x = y
then
ax = ay
Division Property
If
x = y
then
x/a = y/a
(a ≠ 0)
Steps to Solve Linear Equations
Step 1
Variable ko ek side rakhiye.
Step 2
Constants ko dusri side le jaiye.
Step 3
Like terms simplify kijiye.
Step 4
Variable ki value find kijiye.
Solved Examples
Example 1
Solve:
x + 5 = 12
Solution
x = 12 – 5
x = 7
Answer = 7
Example 2
Solve:
2x + 3 = 11
Solution
2x = 11 – 3
2x = 8
x = 4
Answer = 4
Example 3
Solve:
5x – 7 = 18
Solution
5x = 18 + 7
5x = 25
x = 5
Answer = 5
Example 4
Solve:
3x + 8 = 20
Solution
3x = 20 – 8
3x = 12
x = 4
Answer = 4
Example 5
Solve:
x/4 = 5
Solution
x = 20
Answer = 20
Real Life Applications
Linear equations are used in:
- Shopping calculations
- Age problems
- Profit and loss
- Distance and speed
- Business calculations
- Budget planning
Word Problems
Problem 1
A number increased by 8 becomes 25. Find the number.
Solution
Let number = x
x + 8 = 25
x = 17
Answer = 17
Problem 2
Three times a number is 27.
Solution
3x = 27
x = 9
Answer = 9
Key Points
| Variable represents an unknown value. |
| Equation shows equality. |
| Linear equation contains one variable. |
| Highest power of variable is 1. |
| Same operation on both sides keeps equation balanced. |
| Solution means the value of variable that satisfies the equation. |
Abbreviations
| Abbreviation | Full Form |
|---|---|
| LHS | Left Hand Side |
| RHS | Right Hand Side |
| Eqn | Equation |
| Var | Variable |
| Soln | Solution |
| NCERT | National Council of Educational Research and Training |
| HOTS | Higher Order Thinking Skills |
HOTS Questions
Q1
If
3x + 7 = 2x + 15
Find x.
Answer
x = 8
Q2
If
4(x – 2) = 20
Find x.
Answer
x = 7
Q3
The sum of a number and 18 is equal to twice the number.
Find the number.
Answer
18
Q4
A number decreased by 12 equals 20.
Find the number.
Answer
32
Q5
Half of a number plus 5 equals 15.
Find the number.
Answer
20
NCERT Exercise Important Questions & Answers
Question 1
Solve:
x – 2 = 7
Answer
x = 9
Question 2
Solve:
3x = 12
Answer
x = 4
Question 3
Solve:
x/5 = 6
Answer
x = 30
Question 4
Solve:
4x + 3 = 19
Answer
x = 4
Question 5
Solve:
7x – 5 = 30
Answer
x = 5
One Shot Revision
| Variable = Unknown quantity |
| Equation = Equality statement |
| Linear Equation = Variable ki highest power 1 |
| General Form = ax + b = c |
| LHS = Left side |
| RHS = Right side |
| Solve by balancing both sides |
| Check answer by substitution |
| Used in daily life calculations |
🎯 Rising Star Challenge – Tricky Questions for Viewers
In questions ko solve karke comments ya message mein bhejiye. Best answers ko future post mein feature kiya ja sakta hai!
Question 1
A number increased by 15 becomes 42.
What is the number?
Question 2
Five times a number minus 8 equals 27.
Find the number.
Question 3
Solve:
4(x − 3) = 20
Question 4
The present age of a child is x years. After 5 years, his age will be 17 years. Find x.
Question 5 (Challenge Question)
I am thinking of a number.
When I multiply it by 3 and add 7, I get 31.
What is the number?
Conclusion
Linear Equations in One Variable Class 8 Algebra ka foundation chapter hai. Agar aap variables, equations aur balancing method ko achhi tarah samajh lete hain, to aage ke Algebra chapters bahut aasaan ho jate hain. Regular practice aur word problems solve karne se aapki mathematical thinking aur problem-solving skills dono improve hoti hain.
Keep Learning, Keep Growing! 🚀
P Kumar | Rising Star Mindset
www.risingstarmindset.com
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