Algebra - Expressions with Variables

Variable: A symbol, usually a letter like $x$, $y$, or $z$, that represents an unknown numerical value or a value that can change. Visualise a variable as an empty placeholder box or a 'container' where different numbers can be dropped in.

Constant: A value that remains fixed and does not change, such as $5$, $-12$, or $\frac{1}{2}$. In a visual sense, a constant is like a solid brick that always occupies the same space and value regardless of the situation.

Algebraic Expression: A mathematical phrase formed by combining variables and constants using arithmetic operations like addition ($+$), subtraction ($-$), multiplication ($\times$), and division ($\div$). For example, $4x + 7$ is an expression. It can be seen as a set of instructions for a calculation.

Terms: The parts of an expression that are separated by plus ($+$) or minus ($-$) signs. In the expression $3x^2 - 5y + 8$, the terms are $3x^2$, $-5y$, and $8$. Visualise an expression as a train where each carriage is a 'term' connected by operational 'couplers'.

Factors and Coefficients: In a term like $7xy$, the numbers and letters multiplied together ($7$, $x$, and $y$) are called factors. The numerical factor (the number $7$) is specifically called the numerical coefficient. You can imagine a 'factor tree' where the term is the trunk and the individual factors are the branches.

Power/Exponent: When a variable is multiplied by itself multiple times, we use exponential notation. For example, $x \times x$ is $x^2$ (read as $x$ squared) and $x \times x \times x$ is $x^3$ (read as $x$ cubed). Visually, $x^2$ represents the area of a square with side length $x$, while $x^3$ represents the volume of a cube with side length $x$.

Like and Unlike Terms: Like terms are terms that have the same variables raised to the same powers, such as $5x$ and $12x$. Unlike terms have different variables or powers, such as $3x$ and $3y$. Think of like terms as objects of the same shape and size that can be neatly stacked together.

Evaluation by Substitution: This is the process of finding the numerical value of an expression by replacing the variables with given numbers. Visualise 'unplugging' the letter from the expression and 'plugging in' a specific number in its place to solve the arithmetic.