Algebra - Using Expressions Practically

Variables as Unknowns: Variables are symbols, usually letters like $x, y, z, a, b$, or $c$, that represent numbers whose values are not yet fixed. In a practical sense, imagine a box labeled '$x$' where you can place any number of items; the total value changes based on what you put inside.

Forming Algebraic Expressions: We translate real-life situations into mathematical language. For example, if a notebook costs $x$ rupees and a pen costs 5 rupees, the total cost of one notebook and one pen is $x + 5$. Here, the plus sign represents the 'sum' or 'addition' of costs.

Geometric Rules: Algebra allows us to write general formulas for shapes. For a square with a side of length $s$, the perimeter is the sum of all four sides. Visually, this is $s + s + s + s$, which we write concisely as $4s$. Similarly, for an equilateral triangle with side $l$, the perimeter is $3l$.

Matchstick Patterns: We can find rules for patterns made with objects. If one letter 'T' requires 2 matchsticks and two 'T's require 4, we observe a pattern. For $n$ number of 'T's, the number of matchsticks required is $2n$. This visual progression shows how algebra generalizes repeated addition.

Commutative and Distributive Properties: Algebra helps express arithmetic laws. The Commutative Property states $a + b = b + a$, meaning the order of addition doesn't change the sum. The Distributive Property, $a(b + c) = ab + ac$, shows how a number multiplies a group of numbers added together, which is like calculating the area of two adjacent rectangles.

Expressions in Daily Life: We use variables to relate different quantities. If Sarita's current age is $y$ years, then her age 5 years ago was $y - 5$ and her age 10 years from now will be $y + 10$. This allows us to solve problems involving time and growth.

Translating Operations: Key words help build expressions. 'More than' or 'Increased by' signals addition ($+$); 'Less than' or 'Subtracted from' signals subtraction ($-$, note that '$10$ subtracted from $x$' is $x - 10$); 'Times' or 'Product' signals multiplication ($\times$); and 'Divided by' signals division ($\div$).