Quick Revision Notes & Formula Sheet

Class 8 Mathematics — Chapter 2: Linear Equations in One Variable

(एक चर वाले रैखिक समीकरण)

---

1. Basic Concepts & Definitions (मूल अवधारणाएं)

• Variable (चर): A symbol that can take different numerical values. Represented by letters like $x, y, z, a, b, c$.
• Constant (अचर): A quantity having a fixed numerical value. Example: $2, -5, \frac{3}{4}$.
• Algebraic Expression (बीजीय व्यंजक): An expression formed by combining variables and constants using operations like $+$, $-$, $\times$, $\div$.
  • Example: 2x + 5, 3y - 7
• Algebraic Equation (बीजीय समीकरण): An equality statement involving variables and an equal to sign (=).
  • LHS (Left Hand Side) = RHS (Right Hand Side)
  • Example: 2x + 5 = 15
• Linear Equation in One Variable (एक चर वाला रैखिक समीकरण): An equation that contains only one variable and the highest degree (power) of that variable is 1.
  • Standard Form: ax + b = 0 or ax + b = c (where $a \neq 0$)

---

2. Rules for Solving Equations (पक्षांतरण के नियम)

When transposing (shifting) terms from one side of the equal sign (=) to the other:

Operation on LHS	Changes to on RHS	Example
Addition (+)	Subtraction (-)	x + 5 = 12 $\rightarrow$ x = 12 - 5
Subtraction (-)	Addition (+)	x - 3 = 7 $\rightarrow$ x = 7 + 3
Multiplication ($\times$)	Division ($\div$)	4x = 20 $\rightarrow$ x = 20 / 4
Division ($\div$)	Multiplication ($\times$)	x / 3 = 5 $\rightarrow$ x = 5 * 3

---

3. Key Methods & Formulas

Method 1: Balancing Method (संतुलन विधि)

Perform the exact same mathematical operation on both sides of the equation so the equality remains valid.

• Add or subtract the same number on both sides.
• Multiply or divide both sides by the same non-zero number.

Method 2: Transposition Method (पक्षांतरण विधि)

Move constants to one side (usually RHS) and variable terms to the other side (usually LHS) by reversing their operations.

Method 3: Cross-Multiplication Method (वज्र-गुणन विधि)

Used when the equation is in the rational form:

$$\frac{ax + b}{cx + d} = \frac{p}{q}$$

Formula / Rule: $$q \times (ax + b) = p \times (cx + d)$$

---

4. Important Standard Formulas for Word Problems (अनुप्रयोग)

A. Number-Based Problems (संख्या सम्बंधी प्रश्न)

• Consecutive Natural Numbers (क्रमागत संख्याएँ): x, x + 1, x + 2, x + 3, ...
• Consecutive Even/Odd Numbers (क्रमागत सम/विषम संख्याएँ): x, x + 2, x + 4, x + 6, ...
• Two-Digit Number Concept (दो अंकों की संख्या):
  • Let Unit's digit = x, Ten's digit = y
  • Original Number = 10y + x
  • Number after reversing digits = 10x + y

B. Age-Based Problems (आयु सम्बंधी प्रश्न)

• If present age = x years:
  • Age after $n$ years (in future) = x + n years
  • Age $n$ years ago (in past) = x - n years

C. Mensuration / Geometry Formulas (क्षेत्रमिति)

• Perimeter of Rectangle (आयत का परिमाप): 2 * (Length + Breadth)
• Perimeter of Square (वर्ग का परिमाप): 4 * Side
• Perimeter of Triangle (त्रिभुज का परिमाप): Sum of all three sides
• Sum of angles of a triangle: 180°

D. Currency / Coin Problems (सिक्के/नोट सम्बंधी प्रश्न)

• Total Value = Number of notes/coins * Denomination value
  • Example: Value of $x$ notes of ₹50 = 50 * x

---

5. Step-by-Step Guide to Solve Word Problems

1. Read carefully: Identify what is given and what needs to be found.
2. Assume a variable: Represent the unknown quantity as x.
3. Form the equation: Convert the given conditions into a mathematical statement.
4. Solve the equation: Find the value of x using algebraic rules.
5. Verify: Check if the answer satisfies the conditions given in the question.

---

6. Summary Checklist / Quick Tips

• $\checkmark$ Highest power of variable in a linear equation must always be 1.
• $\checkmark$ An equation always has an "=" sign; an algebraic expression does not.
• $\checkmark$ Always simplify both LHS and RHS before transposing terms.
• $\checkmark$ Remember to multiply every term inside brackets when expanding equations like a(bx + c) = abx + ac.