Algebra - Introduction to Equations and their Solutions

An Equation is a mathematical statement that shows two expressions are equal using the equality sign ($=$). Think of an equation as a balanced weighing scale; for the scale to remain level, the total weight on the Left Hand Side (LHS) must be exactly the same as the total weight on the Right Hand Side (RHS).

Variables and Constants are the basic components of an algebraic equation. A variable is a symbol (usually a letter like $x$, $y$, or $z$) that represents an unknown number, while a constant is a fixed numerical value like $5, -10,$ or $0.5$.

A Linear Equation in One Variable is an equation where the highest power of the variable is $1$. For example, $3x + 4 = 10$ is a linear equation. Visually, the solution to such an equation represents a single specific point on a number line.

The Solution or Root of an equation is the value of the variable that makes the LHS equal to the RHS. If you substitute the solution back into the equation, the mathematical statement becomes true (e.g., in $x + 2 = 5$, the solution is $x = 3$ because $3 + 2 = 5$).

The Balancing Method is a fundamental rule stating that to maintain equality, any operation performed on one side of the equation must also be performed on the other side. Imagine adding a $2$ kg weight to the left pan of a balanced scale; you must add $2$ kg to the right pan to keep it from tipping.

Transposition is the technique of moving a term from one side of the equals sign to the other by changing its sign or operation. When a term moves across the 'bridge' of the $=$ sign, addition becomes subtraction ($+$ to $-$), subtraction becomes addition ($-$ to $+$), multiplication becomes division ($imes$ to $\div$), and division becomes multiplication ($\div$ to $imes$).