Algebra - Introduction to Algebra and Variables

A variable is a quantity that does not have a fixed value and is represented by letters like $x, y, z, l, m, n$. Unlike constants (fixed numbers like $5$ or $100$), variables allow us to write general rules. Visually, think of a variable as an empty box or a placeholder where any number can be placed.

Algebraic rules can be derived from patterns, such as matchstick shapes. For example, if one letter 'L' requires $2$ matchsticks, then $n$ letters of 'L' require $2 \times n = 2n$ sticks. Visually, as you add more shapes to a row, the total number of sticks increases by a fixed amount for every new unit added.

Variables are used to generalize geometry formulas. For a square with side length $s$, the perimeter is the sum of its four equal sides. Visually, a square consists of four equal line segments, so we write the rule as $P = 4s$. For a rectangle with length $l$ and breadth $b$, the perimeter is $2(l + b)$.

In arithmetic, variables help express properties of numbers concisely. The Commutative Property of addition states $a + b = b + a$. Visually, if you combine a blue strip of length $a$ with a red strip of length $b$, the total length is the same regardless of which strip is placed first.

The Distributive Property of multiplication over addition is expressed as $a \times (b + c) = (a \times b) + (a \times c)$. This can be visualized by finding the area of a large rectangle by splitting it into two smaller rectangles and adding their individual areas.

Algebraic expressions are formed by combining variables and constants using operators like $+$, $-$, $\times$, and $\div$. For example, $y + 10$ means $10$ is added to $y$. Visually, on a number line, $y + 10$ represents a point $10$ units to the right of $y$.

An equation is a condition on a variable showing that two expressions are equal, separated by an '=' sign. It has a Left Hand Side (LHS) and a Right Hand Side (RHS). Visually, an equation is like a balanced weighing scale; the scale remains level only when the values on both sides are exactly equal.

The solution of an equation is the specific value of the variable that makes the LHS equal to the RHS. For example, in the equation $x + 5 = 12$, the solution is $x = 7$ because $7 + 5 = 12$ makes the equation true.