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Algebra I

Mathematics

Translate real-world situations into symbolic equations and solve them using variables, expressions, and systematic methods. Master linear functions, systems of equations, polynomials, and quadratics—each concept building on the last to develop algebraic reasoning and problem-solving skills.

What you'll learn

Solving Multi-Step Linear EquationsThe process of isolating a variable in an equation that requires multiple inverse operations, typically involving distribution, combining like terms, moving terms, and division.
Variables and Algebraic ExpressionsLetters that represent unknown or changeable numbers, combined with operations and constants to form expressions that describe relationships without using an equals sign.
The Coordinate Plane and RelationsA flat grid formed by two perpendicular number lines (the x-axis and y-axis) that allows you to locate any point using two numbers called an ordered pair.
Properties of Real NumbersFormal rules that justify rearranging and regrouping numbers, including the commutative properties that allow reordering and the associative properties that allow regrouping in addition and multiplication.
Slope of a LineA numerical measure of the steepness and direction of a line, calculated as the ratio of vertical change to horizontal change between any two points on the line.
Function Notation and EvaluationA function is a rule that assigns exactly one output to each input, and the notation f(x) represents the output when x is the input to function f.
Slope-Intercept FormA linear equation written as y = mx + b where m is the slope and b is the y-intercept, allowing both the steepness and starting point of a line to be read directly from the equation.
Solving and Graphing Linear InequalitiesA technique for finding all values of a variable that satisfy a comparison statement using symbols like <, >, ≤, or ≥, with the key difference that multiplying or dividing by a negative number requires flipping the inequality sign.
Graphing ParabolasThe method for sketching a quadratic function by identifying its direction of opening, axis of symmetry, vertex, and intercepts.
Introduction to Quadratic FunctionsA quadratic function is a polynomial function of degree 2 that can be written as f(x) = ax² + bx + c where a ≠ 0, and its graph forms a U-shaped or inverted U-shaped curve called a parabola.
Factoring TrinomialsA trinomial expression is factored by finding two numbers that multiply to the constant term and add to the coefficient of the middle term, reversing the process of binomial multiplication.
Solving Quadratics by FactoringA method for solving quadratic equations by rewriting the equation in standard form, factoring the quadratic expression, and applying the Zero Product Property to find solutions.
Intercepts of a LineThe points where a line crosses the x-axis (where y equals zero) or the y-axis (where x equals zero).
Absolute Value EquationsEquations involving absolute value split into two cases based on the definition that absolute value represents distance from zero on the number line.
Graphing Linear SystemsA system of linear equations is solved by graphing both equations on the same coordinate plane and finding the point where the two lines intersect, which represents the values that satisfy both equations simultaneously.
Systems of Linear InequalitiesA set of two or more linear inequalities in two variables whose solution is the overlapping region on the coordinate plane where all inequalities are satisfied simultaneously.
Solving Quadratics by Completing the SquareA method for solving quadratic equations by transforming an expression into a perfect square trinomial, which works for every quadratic equation regardless of whether it factors with integers.
The Quadratic FormulaA universal method that solves any quadratic equation in standard form by substituting the coefficients into the formula x = (-b ± √(b² - 4ac)) / (2a).
Introduction to PolynomialsAn algebraic expression formed by adding or subtracting terms where each term is a product of numbers and variables raised to whole-number exponents.
Solving Radical EquationsThe process of solving equations where a variable appears inside a root by isolating the radical, squaring both sides, solving the resulting equation, and checking for extraneous solutions.
Exponent RulesAlgebraic shortcuts for combining powers with the same base during multiplication, division, and exponentiation without writing out every repeated factor.
Greatest Common Factor FactoringThe process of rewriting a polynomial as a product by extracting the largest number and highest powers of variables that divide evenly into every term.
Solving Systems by EliminationA method for solving systems of linear equations by adding or subtracting equations so that one variable cancels out, leaving a single equation that can be solved directly.
Multiplying PolynomialsMultiplying polynomials involves applying the distributive property repeatedly to combine each term of one polynomial with every term of another, with special procedures like FOIL for binomials.
Simplifying Radical ExpressionsThe process of rewriting a radical expression so that the radicand contains no remaining perfect square factors.
Translating Word Problems into EquationsThe process of converting a real-world situation described in words into an algebraic equation using variables to represent unknown quantities.
Standard Form of a LineThe equation form Ax + By = C where A, B, and C are constants, useful for quickly finding intercepts and modeling real-world situations with two quantities that trade off.
Interpreting and Comparing FunctionsThe skill of extracting key information like slope and starting value from equations, tables, or graphs to analyze how functions behave relative to one another.
Simplifying Rational ExpressionsThe process of reducing a fraction with polynomial numerator and denominator by factoring both completely and canceling common factors.
Lines of Best FitA line of best fit is a straight line drawn through a scatter plot to model the overall linear relationship between two variables, allowing for predictions and numerical description of the trend even when the line does not pass through individual data points.
Classifying Systems of EquationsThe process of determining whether a system of two linear equations has one solution, no solution, or infinitely many solutions by comparing the slopes and y-intercepts of their graphs.
Arithmetic SequencesAn ordered list of numbers where each term is obtained by adding the same fixed number (the common difference) to the previous term.
Operations on Rational ExpressionsRational expressions are fractions containing polynomials in the numerator, denominator, or both, and they can be simplified, multiplied, divided, added, and subtracted using the same rules as numeric fractions along with factoring techniques.
Factoring Special PatternsQuadratic expressions that follow predictable structures, like the difference of squares, can be factored by recognizing the pattern rather than using trial-and-error.
Factoring by GroupingA technique for factoring four-term polynomials by splitting them into pairs, factoring each pair separately, and pulling out the common binomial factor that appears in both.
Solving Systems by SubstitutionAn algebraic method for solving systems of equations by expressing one variable in terms of another and replacing it in the second equation to create a one-variable equation.
Compound InequalitiesAn inequality statement joining two conditions with "and" or "or", where "and" requires both conditions true simultaneously and "or" requires at least one condition true.
Adding and Subtracting PolynomialsCombining polynomials by adding or subtracting the coefficients of like terms while keeping their variable parts unchanged.
Point-Slope FormPoint-slope form is a way to write the equation of a line using a known point (x₁, y₁) on the line and its slope m, written as y − y₁ = m(x − x₁).
Absolute Value InequalitiesInequalities involving absolute value follow two patterns where less-than comparisons produce bounded intervals and greater-than comparisons produce two separate regions.
Scatter Plots and CorrelationA graph that displays the relationship between two numeric variables by plotting each pair of data values as a point on the coordinate plane.
Equations with Variables on Both SidesLinear equations in which the variable appears in terms on multiple sides, solved by moving all variable terms to one side before isolating the variable.
Writing Equations of Parallel and Perpendicular LinesTwo lines are parallel when they have the same slope, and perpendicular when their slopes are negative reciprocals of each other.
Choosing a Quadratic Solving MethodThe skill of recognizing which of three algebraic techniques—factoring, completing the square, or the quadratic formula—is fastest and least error-prone for a given quadratic equation.

About this course

Welcome to Algebra I

Up to this point, your math classes have mostly worked with numbers you already know: adding fractions, computing percentages, simplifying expressions with a single unknown value. Algebra I marks a shift. Instead of only working with known numbers, you'll build tools to work with quantities that are unknown, changing, or dependent on other quantities. This overview is your map of the territory — a preview of the major ideas you'll meet in this course and how they connect to each other.

Why Algebra Exists

Imagine you're saving money for a bike that costs $180. You already have $45, and you save $15 per week. How many weeks until you can afford it? You could guess and check, but algebra gives you a systematic way to represent this situation with symbols — $45 + 15w = 180$ — and then solve for $w$ using reliable rules. That's the core purpose of algebra: turning real situations into symbolic models, then manipulating those symbols to find answers that would be hard to find by guessing.

This pattern — translate a situation into an equation, then solve it — will repeat throughout the entire course, across every topic from simple equations to quadratics to systems of equations.

The Big Ideas of the Course

1. Variables and Expressions. Everything starts with the idea of a variable: a letter that stands in for a number. You'll learn to write expressions using variables, evaluate them for given values, and simplify them using order of operations and the properties of real numbers (commutative, associative, distributive). These are the grammar rules of algebra — the manipulations you're allowed to perform and why they're valid.

2. Equations and Inequalities. Once you can write expressions, the next step is solving equations — finding the value(s) of a variable that make a statement true. You'll start with multi-step equations, then equations with variables on both sides, then absolute value equations (which split into two cases), and inequalities (where the solution is often a range of values rather than a single number, and where compound inequalities join two conditions with "and" or "or").

3. Systems of Equations. Real situations often involve more than one unknown and more than one relationship at once — for example, finding the price of two different items given two purchase receipts. Systems of linear equations let you solve for multiple unknowns using three methods: graphing (finding where lines intersect), substitution, and elimination. Each method has situations where it works best.

4. Linear Functions and Graphs. A function is a relationship where each input has exactly one output. Linear functions — the simplest and most common type — are the algebraic backbone of the course. You'll learn to graph them, find their slope (rate of change) and intercepts, write them in different forms (slope-intercept, point-slope, standard form), and connect them to arithmetic sequences, where a constant difference between terms mirrors the constant slope of a line.

5. Polynomials. As expressions grow more complex — involving terms like $x^2$ , $x^3$ , or products of multiple variables — you need exponent rules to manage them and operations (addition, subtraction, multiplication) to combine them. This sets the stage for factoring, which reverses multiplication: breaking an expression like $x^2 + 5x + 6$ back into $(x+2)(x+3)$ .

6. Quadratic Equations and Functions. Quadratics — equations involving $x^2$ — introduce curved graphs called parabolas and require new solving techniques: factoring, completing the square, and the quadratic formula. This is often considered the centerpiece of Algebra I, tying together everything from factoring to graphing.

7. Radicals and Rational Expressions. These topics introduce square roots and fractions containing variables, previewing techniques you'll expand on in Algebra II. You'll learn to simplify radical expressions, solve radical equations (watching for extraneous solutions), and perform basic operations on rational expressions.

8. Function Notation and Data Modeling. Toward the end of the course, you'll formalize the function idea using notation like $f(x)$ , compare linear and nonlinear functions, and use scatter plots and lines of best fit to model real data — a bridge toward statistics and higher-level modeling.

How These Pieces Fit Together

Figure: How the major units of Algebra I build on one another, from basic expressions to advanced modeling topics.

Notice the pattern: expressions support equations, equations support systems and functions, and functions support polynomials, which support quadratics. Nothing here is isolated — each new skill leans on the ones before it.

What This Course Does Not Cover

To keep your focus sharp, this course stops short of matrices, logarithms, complex numbers, conic sections, and trigonometry — these belong to Algebra II and beyond. What you learn here is the essential foundation that makes those later topics possible.

How to Use This Course

Algebra rewards patience and practice more than natural talent. Every concept builds on the last, so if something feels shaky, it's worth pausing to strengthen it before moving on. Expect worked examples, step-by-step explanations, and practice problems that increase gradually in difficulty. By the end, you'll be able to take a word problem — about money, distance, mixtures, or growth — and confidently turn it into an equation, solve it, and interpret what the answer means.