Algebra 1

Unit 1

The Real Number System & Order of Operations

Evaluate expressions using PEMDAS: Parentheses → Exponents → Multiplication/Division (left→right) → Addition/Subtraction (left→right).

P · E · M/D · A/S

Worked Example

$3 + 4 \times 2 - 6 \div 3 = 3 + 8 - 2 = \mathbf{9}$

Unit 2

Expressions, Properties & Simplifying

Distributive: $a(b+c) = ab + ac$ Combining like terms: same variable & exponent.

$3(2x - 5) = 6x - 15$

Worked Example

$5x + 3 - 2x + 7 = (5-2)x + (3+7) = \mathbf{3x + 10}$

Unit 3

Solving Linear Equations

Isolate the variable by performing inverse operations on both sides. Check your answer by substituting back.

$2x + 5 = 13 \;\Rightarrow\; 2x = 8 \;\Rightarrow\; x = 4$

Variable on both sides

$3x - 2 = x + 8 \;\Rightarrow\; 2x = 10 \;\Rightarrow\; x = 5$

Unit 4

Linear Functions & Graphing

Slope: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ Slope-intercept: $y = mx + b$

$y - y_1 = m(x - x_1)$ ← Point-slope form

Worked Example

Points $(2,3)$ and $(6,11)$: $m = \frac{11-3}{6-2} = 2$. Through $(1,4)$ with $m=3$: $y = 3x+1$

Unit 5

Systems of Linear Equations

Substitution: solve one equation for a variable, substitute into the other.

Elimination: add or subtract equations to cancel one variable.

$\begin{cases} y = 2x+1 \\ 3x+y=16 \end{cases} \Rightarrow (3,\,7)$

Unit 6

Polynomials & Factoring

FOIL: $(a+b)(c+d) = ac + ad + bc + bd$

Factor $ax^2+bx+c$: find two numbers that multiply to $ac$ and add to $b$.

$(x+3)(x-2) = x^2 + x - 6$ | $x^2-16 = (x-4)(x+4)$

Unit 7

Quadratic Equations

Factoring: set each factor = 0. Quadratic Formula: $x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$

$x^2 - 7x + 10 = 0 \Rightarrow (x-2)(x-5)=0 \Rightarrow x=2,\,5$

Unit 8

Functions, Inequalities & Sequences

Function notation: $f(x)$ — substitute the value for $x$.

Inequalities: flip the sign when multiplying/dividing by a negative.

Arithmetic sequence: $a_n = a_1 + (n-1)d$

$|2x-3|=7 \Rightarrow x=5$ or $x=-2$

Key Concepts to Master

The Real Number System & Order of Operations

Expressions, Properties & Simplifying

Solving Linear Equations

Linear Functions & Graphing

Systems of Linear Equations

Polynomials & Factoring

Quadratic Equations

Functions, Inequalities & Sequences