Quick Revision Notes: Class 6 Mathematics

Chapter: Algebra (बीजगणित)

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### 1. What is Algebra? (बीजगणित क्या है?)

• Definition: Algebra is a branch of mathematics in which we use letters or symbols in place of unknown quantities (numbers).
• Purpose: It helps us to express general rules, formulas, and solve puzzles or unknown values in arithmetic.
• Arabic Origin: The word 'Algebra' comes from the Arabic book Al-Jabr w'al-muqabala.

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### 2. Key Concepts & Terms (मुख्य अवधारणाएँ)

• Variable (चर):

  • A quantity that can change or take different numerical values.
  • Usually denoted by English letters like $x, y, z, p, q, n$, etc.
  • Example: In $x + 5$, $x$ is a variable.

• Constant (अचर):

  • A quantity that has a fixed numerical value.
  • Example: $3, -7, \frac{1}{2}, 0$ are constants.

• Literal Numbers (बीजीय संख्याएँ): Letters used to represent numbers.

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### 3. Use of Variables in Common Rules (सामान्य नियमों में चरों का प्रयोग)

A. Rules of Geometry (रेखाyगणित के नियम)

1. Perimeter of a Square (वर्ग का परिमाप):
   • $P = 4 \times s$ (where $s$ is the side of the square)
2. Perimeter of a Rectangle (आयताकार का परिमाप):
   • $P = 2 \times (l + b)$ (where $l$ is length and $b$ is breadth)
3. Area of a Square (वर्ग का क्षेत्रफल):
   • $A = s \times s = s^2$
4. Area of a Rectangle (आयता का क्षेत्रफल):
   • $A = l \times b$

B. Rules of Arithmetic (अंकगणित के नियम)

1. Commutativity of Addition (योग की क्रमविनिमयता):
   • $a + b = b + a$ (For any numbers $a$ and $b$)
2. Commutativity of Multiplication (गुणन की क्रमविनिमयता):
   • $a \times b = b \times a$
3. Associativity of Addition (योग की साहचर्यता):
   • $(a + b) + c = a + (b + c)$
4. Distributivity of Multiplication over Addition (वितरण गुण):
   • $a \times (b + c) = (a \times b) + (a \times c)$

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### 4. Expressions with Variables (चरों वाले व्यंजक)

Operations like addition, subtraction, multiplication, and division are combined with constants and variables to form algebraic expressions.

Statement (कथन)	Algebraic Expression (बीजीय व्यंजक)
$x$ increased by $5$ ($x$ में $5$ जोड़ना)	$x + 5$
$7$ subtracted from $y$ ($y$ में से $7$ घटाना)	$y - 7$
$4$ times $p$ ($p$ का $4$ गुना)	$4p$ (or $4 \times p$)
$m$ divided by $3$ ($m$ को $3$ से भाग देना)	$\frac{m}{3}$
$2$ subtracted from $3$ times $x$ ($x$ के $3$ गुने में से $2$ घटाना)	$3x - 2$

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### 5. What is an Equation? (समीकरण क्या है?)

• Definition: An equation is a condition on a variable. It has an equality sign ($=$) between two sides.

• Parts of an Equation:

  • L.H.S. (Left Hand Side): Expression on the left of the '=' sign.
  • R.H.S. (Right Hand Side): Expression on the right of the '=' sign.

• Condition: An equation holds true only for a certain value of the variable. This value is called the solution or root of the equation.

• Example:

  • $4x + 5 = 25$ is an equation.
  • Here, L.H.S. $= 4x + 5$ and R.H.S. $= 25$.

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### 6. Solutions of an Equation (समीकरण का हल)

• Trial and Error Method (प्रयोजन और त्रुटि विधि):

  • Substituting different values for the variable in the equation until L.H.S. equals R.H.S.

• Example: Solve $x + 3 = 8$

  • Put $x = 1 \rightarrow 1 + 3 = 4 \neq 8$
  • Put $x = 3 \rightarrow 3 + 3 = 6 \neq 8$
  • Put $x = 5 \rightarrow 5 + 3 = 8 = \text{R.H.S.}$ (Correct!)
  • Therefore, $x = 5$ is the solution of the equation.