Algebra - Introduction to Algebra: Variables and Constants

Constants: A constant is a fixed numerical value that does not change regardless of the situation. For example, in the expression $x + 7$, the number $7$ is a constant. Visually, imagine a solid stone block that never changes size, representing fixed values like $4$, $-12$, or $0.5$.

Variables: A variable is a symbol, usually a lowercase letter like $x$, $y$, $a$, or $b$, used to represent an unknown quantity or a value that can vary. Visually, think of a variable as an empty container or a 'mystery box' where different numbers can be placed inside depending on the problem.

Algebraic Expressions: An algebraic expression is a mathematical phrase created by combining variables, constants, and operations such as addition ($+$), subtraction ($-$), multiplication ($\times$), and division ($\div$). For example, $4y - 3$ is an expression. Visually, this can be seen as a chain or a flow diagram connecting numbers and letters.

Terms: Terms are the individual parts of an algebraic expression that are separated by the plus ($+$) or minus ($-$) signs. In the expression $2x + 5y - 9$, the terms are $2x$, $5y$, and $-9$. Visually, you can imagine the expression as a train where each carriage represents a distinct term.

Coefficients: In a term that contains a variable, the numerical factor is called the coefficient. In the term $8k$, the number $8$ is the coefficient of $k$. Visually, the coefficient is like a multiplier 'stuck' to the front of the variable, telling you how many copies of that variable you have.

Translating Word Phrases: Algebra acts as a bridge between English and Mathematics. Phrases like 'increased by' or 'sum' translate to $+$, 'decreased by' or 'less than' translate to $-$, and 'product' translates to $\times$. Visually, imagine a translation dictionary where words point to their specific mathematical symbols.

Evaluating Expressions: This involves finding the numerical value of an expression by substituting specific numbers for the variables. If $p = 5$, then the expression $p + 10$ becomes $5 + 10 = 15$. Visually, this is the process of removing the 'variable box' and dropping in a specific number block to calculate a final result.