Class 6 Algebra Basics

Cheatsheet Content

### Introduction to Algebra Algebra is a branch of mathematics that uses letters (variables) to represent numbers. These letters help us to write rules and solve problems where some numbers are unknown. - **Variables:** Letters like $x, y, z, a, b, c$ that stand for unknown numbers. - **Constants:** Numbers that have a fixed value, like $5, 10, -3$. - **Expressions:** Combinations of variables, constants, and mathematical operations. - Example: $x + 5$, $3y - 2$, $2a + 3b$ - **Equations:** Statements that show two expressions are equal. They always have an '=' sign. - Example: $x + 5 = 10$, $3y - 2 = 7$ ### Basic Operations with Variables The same rules of addition, subtraction, multiplication, and division apply to variables. - **Addition:** - $x + x = 2x$ - $x + y$ (cannot be simplified if $x$ and $y$ are different variables) - **Subtraction:** - $y - y = 0$ - $3a - a = 2a$ - **Multiplication:** - $x \times y = xy$ - $2 \times a = 2a$ - $a \times a = a^2$ (read as 'a squared') - **Division:** - $x \div y = \frac{x}{y}$ - $4a \div 2 = \frac{4a}{2} = 2a$ ### Forming Expressions Translating word problems into algebraic expressions. | Word Phrase | Algebraic Expression | |-------------------------|----------------------| | 5 more than $x$ | $x + 5$ | | 10 less than $y$ | $y - 10$ | | Twice $a$ | $2a$ | | One-third of $b$ | $\frac{b}{3}$ | | Product of $x$ and $y$ | $xy$ | | Quotient of $p$ and $q$ | $\frac{p}{q}$ | | 7 added to $m$ | $m + 7$ | | $k$ decreased by 4 | $k - 4$ | ### Solving Simple Equations To solve an equation means to find the value of the variable that makes the equation true. - **Rule 1: Addition/Subtraction Principle** - If you add or subtract the same number from both sides of an equation, the equation remains balanced. - Example: $x - 3 = 7$ - Add 3 to both sides: $x - 3 + 3 = 7 + 3$ - $x = 10$ - **Rule 2: Multiplication/Division Principle** - If you multiply or divide both sides of an equation by the same non-zero number, the equation remains balanced. - Example: $2x = 12$ - Divide both sides by 2: $\frac{2x}{2} = \frac{12}{2}$ - $x = 6$ **Steps to solve:** 1. Isolate the variable on one side of the equation. 2. Use inverse operations to move constants to the other side. **Example 1:** Solve $x + 4 = 9$ - Subtract 4 from both sides: $x + 4 - 4 = 9 - 4$ - $x = 5$ **Example 2:** Solve $\frac{y}{3} = 5$ - Multiply both sides by 3: $\frac{y}{3} \times 3 = 5 \times 3$ - $y = 15$ ### Patterns and Rules Algebra helps us describe patterns using general rules. - **Matchstick Patterns:** - If 3 matchsticks form a triangle, then $n$ triangles will need $3 \times n$ matchsticks ($3n$). - **Number Patterns:** - Sequence: 2, 4, 6, 8, ... - Rule: Each number is 2 times its position. So, the $n$-th term is $2n$. - **Perimeter Formulas:** - Perimeter of a square with side $s$: $P = 4s$ - Perimeter of a rectangle with length $l$ and width $w$: $P = 2(l + w)$