Algebra 1 — Variables and Expressions (Lesson 1) Summary & Study Notes

🧩 What is a Variable?

A variable is a symbol (usually a lowercase letter) that acts as a placeholder for an unknown quantity. Examples you will see often are $x$, $y$, and $z$, and each can take different numerical values in different situations.

🧮 Modeling with Variables

We use variables to model real situations. For example: Mark earns $63 + x$ dollars per day where $x$ is the unknown amount of tips. If $x=7$, substitute and evaluate to find $63 + 7 = 70$. If $x=9$, then $63 + 9 = 72$.

Another example: Jason buys two gallons of milk at an unknown price $y$ dollars per gallon. The total cost is written as $2y$. If $y=1$, then $2y=2$; if $y=3$, then $2y=6$.

🔢 Terms, Coefficients, and Constants

A term is a single number, a variable, or a product of numbers and variables (for example, $4x$, $9y$, $24xyz$, or $7$). The number multiplying the variable(s) is the coefficient (e.g., in $4x^2z$ the coefficient is $4$). A constant is a number on its own that does not change value, such as the $7$ in $5x + 7$.

✂️ Algebraic Expressions

An algebraic expression is one or more terms separated by plus or minus signs. For example, $5x + 2y - 4z$ has three terms: $5x$, $2y$, and $-4z$. The value of an algebraic expression changes when the values of its variables change.

🔁 Substitution and Evaluation

To evaluate an expression, replace each variable with a given number and compute. For $2x + 3$ with $x=4$, substitute to get $2(4) + 3 = 8 + 3 = 11$. If $x=-2$, then $2(-2) + 3 = -4 + 3 = -1$.

➗ Distributive Property and Simplifying

The distributive property lets you multiply a factor across terms inside parentheses: $a(b + c) = ab + ac$. For instance, $3(x - 4)$ becomes $3x - 12$. In $4(x - 7) + 2$, distribute to get $4x - 28 + 2$, then combine constants to get $4x - 26$.

≡ Like Terms and Combining

Like terms have the exact same variable part (same letters and same exponents). For example, $5x$ and $3x$ are like terms; $2x$ and $3x^3$ are not. You can combine like terms by adding or subtracting their coefficients, but you cannot combine unlike terms.

🔄 Order of Operations and Properties

When evaluating expressions, follow the order of operations (evaluate parentheses and exponents, then multiplication/division, then addition/subtraction). Use properties such as the commutative property (order doesn’t matter for addition and multiplication) and the associative property (grouping doesn’t affect the result) to rearrange or simplify expressions safely.

🧰 Where This Leads (Factoring)

Simplifying sometimes works in reverse by identifying a common factor and factoring it out; this is the opposite of distribution and is called factoring. Factoring will be explored further in later lessons.

✅ Practical Tips

• Always identify terms by locating plus and minus separators.
• When a number is written next to a variable like $2y$, it means multiplication: $2y = 2\times y$.
• Substitute values carefully and follow the order of operations when evaluating.

These core ideas—variables, terms, coefficients, constants, substitution, distribution, and like terms—form the foundation of Algebra 1 and will repeat throughout the course.