🔤 Algebra — Reference
Essential concepts and definitions for working with algebra in MATHCOUNTS.
Variables and Expressions
Basic Concepts
Variable: A letter representing an unknown number (e.g., $x$, $y$, $z$) Expression: A mathematical phrase with variables and numbers (e.g., $2x + 3$) Term: A single number or variable, or product of numbers and variables Coefficient: The number multiplying a variable (e.g., in $3x$, the coefficient is 3)
Types of Expressions
Monomial: Single term (e.g., $3x$, $5$) Binomial: Two terms (e.g., $2x + 3$) Trinomial: Three terms (e.g., $x^2 + 2x + 1$) Polynomial: Multiple terms (e.g., $x^3 + 2x^2 + x + 5$)
Like Terms
Definition: Terms with the same variables and exponents Examples: $3x$ and $5x$ are like terms; $3x$ and $3x^2$ are not Combining: Add or subtract coefficients: $3x + 5x = 8x$
Linear Equations
Basic Form
Standard form: $ax + b = 0$ where $a \neq 0$ Slope-intercept form: $y = mx + b$ where $m$ is slope, $b$ is y-intercept Point-slope form: $y - y_1 = m(x - x_1)$ where $(x_1, y_1)$ is a point
Solving Linear Equations
Method 1: Isolate the variable
- Add/subtract to both sides
- Multiply/divide both sides
- Check solution
Method 2: Use properties of equality
- Addition property: If $a = b$, then $a + c = b + c$
- Multiplication property: If $a = b$, then $ac = bc$
Example: Solve $2x + 3 = 11$
- Subtract 3: $2x = 8$
- Divide by 2: $x = 4$
- Check: $2(4) + 3 = 11$ ✓
Systems of Equations
Two Variables
Standard form: $ax + by = c$ Slope-intercept form: $y = mx + b$
Solving Methods
Substitution: Solve one equation for one variable, substitute into other Elimination: Add/subtract equations to eliminate one variable Graphing: Plot both equations, find intersection point
Example: Solve $\begin{cases} x + y = 5 \ 2x - y = 1 \end{cases}$
- Elimination: Add equations: $3x = 6$, so $x = 2$
- Substitute: $2 + y = 5$, so $y = 3$
- Solution: $(2, 3)$
Quadratic Equations
Basic Form
Standard form: $ax^2 + bx + c = 0$ where $a \neq 0$ Vertex form: $y = a(x - h)^2 + k$ where $(h, k)$ is vertex Factored form: $y = a(x - r_1)(x - r_2)$ where $r_1, r_2$ are roots
Solving Methods
Factoring: Factor into $(x - r_1)(x - r_2) = 0$ Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Completing the square: Rewrite as $(x - h)^2 = k$
Example: Solve $x^2 - 5x + 6 = 0$
- Factoring: $(x - 2)(x - 3) = 0$
- Solutions: $x = 2$ or $x = 3$
Inequalities
Basic Concepts
Inequality symbols: $<$, $>$, $\leq$, $\geq$, $\neq$ Solution: All values that make the inequality true Graph: Number line or coordinate plane
Solving Inequalities
Method: Similar to equations, but flip inequality when multiplying/dividing by negative Example: Solve $2x + 3 < 11$
- Subtract 3: $2x < 8$
- Divide by 2: $x < 4$
- Graph: Open circle at 4, arrow left
Compound Inequalities
And: $a < x < b$ (both conditions must be true) Or: $x < a$ or $x > b$ (either condition can be true)
Functions
Basic Concepts
Function: A rule that assigns exactly one output to each input Domain: Set of all possible inputs Range: Set of all possible outputs Notation: $f(x) = 2x + 3$ means “f of x equals 2x plus 3”
Function Types
Linear: $f(x) = mx + b$ Quadratic: $f(x) = ax^2 + bx + c$ Absolute value: $f(x) = |x|$ Square root: $f(x) = \sqrt{x}$
Function Operations
Evaluation: Substitute value for variable Composition: $(f \circ g)(x) = f(g(x))$ Inverse: $f^{-1}(x)$ undoes $f(x)$
Patterns and Sequences
Arithmetic Sequences
Definition: Sequence where difference between consecutive terms is constant Formula: $a_n = a_1 + (n-1)d$ where $d$ is common difference Example: 2, 5, 8, 11, … has $d = 3$
Geometric Sequences
Definition: Sequence where ratio between consecutive terms is constant Formula: $a_n = a_1 \cdot r^{n-1}$ where $r$ is common ratio Example: 2, 6, 18, 54, … has $r = 3$
Fibonacci Sequence
Definition: Each term is sum of previous two terms Formula: $F_n = F_{n-1} + F_{n-2}$ with $F_1 = 1$, $F_2 = 1$ Example: 1, 1, 2, 3, 5, 8, 13, …
Word Problems
Translation
Key words:
- “is” or “equals” → $=$
- “more than” or “increased by” → $+$
- “less than” or “decreased by” → $-$
- “times” or “of” → $\times$
- “divided by” → $\div$
Example: “Five more than twice a number is 13”
- Translation: $2x + 5 = 13$
- Solution: $x = 4$
Common Problem Types
Age problems: Use variables for ages Distance problems: Use $d = rt$ formula Mixture problems: Use weighted averages Work problems: Use combined rates
Properties and Rules
Commutative Properties
Addition: $a + b = b + a$ Multiplication: $ab = ba$
Associative Properties
Addition: $(a + b) + c = a + (b + c)$ Multiplication: $(ab)c = a(bc)$
Distributive Property
Multiplication over addition: $a(b + c) = ab + ac$ Multiplication over subtraction: $a(b - c) = ab - ac$
Order of Operations
PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction Example: $2 + 3 \times 4^2 = 2 + 3 \times 16 = 2 + 48 = 50$
Common Mistakes
Sign Errors
Mistake: $-(x + 3) = -x + 3$ Fix: $-(x + 3) = -x - 3$
Distribution Errors
Mistake: $2(x + 3) = 2x + 3$ Fix: $2(x + 3) = 2x + 6$
Inequality Errors
Mistake: When multiplying by negative, not flipping inequality Fix: Always flip inequality when multiplying/dividing by negative