Table of Contents
Overview
Algebra I is your first sustained experience with symbolic algebra. In earlier parts of the course, you met variables, simple equations, and the coordinate plane. Algebra I takes those ideas and turns them into a toolkit for describing and solving a wide variety of problems.
In this chapter, you will not yet go into the detailed techniques listed in the later Algebra I subsections (like “Slope” or “Factoring method”). Instead, you will:
- See what kinds of problems Algebra I is designed to handle.
- Understand the main types of objects studied: linear expressions, linear equations, polynomials, and quadratic equations.
- Get a sense of how Algebra I connects arithmetic to geometry (through graphs) and to problem solving (through word problems).
- Learn what it means to “think algebraically,” beyond just calculating.
Algebra I lays the foundation for all later algebra (including Algebra II and beyond), for coordinate geometry, and for much of calculus and statistics.
The Goals of Algebra I
Algebra I has a few central goals:
- Generalize arithmetic using symbols.
Instead of only solving specific numerical problems like + 5$, you learn to work with general expressions like $x + 5$ or x - 3$ that stand for many possible numbers at once. - Solve equations systematically.
You learn methods for finding values of a variable that make an equation true, such as x + 3 = 11$ or x - 5 = 2x + 7$, and later more complex polynomial equations. - Connect algebra to geometry.
You see how equations describe graphs on the coordinate plane. For example, each linear equation in two variables corresponds to a line. - Model real-world situations.
Word problems are translated into equations or systems of equations. Algebra becomes a language for expressing relationships such as “cost equals rate times time” or “distance equals speed times time.” - Prepare for more advanced mathematics.
Concepts like functions, polynomials, and quadratic equations are the building blocks of later topics like exponential functions, trigonometry, and calculus.
Key Types of Objects in Algebra I
Within Algebra I, you focus on a particular collection of algebraic objects. Later chapters in Algebra I will treat each of these in depth, but here we highlight what they are and why they matter.
Linear Expressions and Linear Equations
A linear expression in one variable is an expression that can be written in the form
$$
ax + b,
$$
where $a$ and $b$ are numbers (constants) and $x$ is a variable. No powers of $x$ other than $x^1$ appear, and $x$ is not in a denominator.
Examples of linear expressions:
- $3x - 2$
- $-5x + 7$
- $\frac{1}{2}x - 4$
A linear equation in one variable sets a linear expression equal to a constant or another linear expression, for example:
- $3x - 2 = 10$
- $2x + 5 = 7x - 3$
In Algebra I you learn systematic ways to solve such equations—ways that work for whole families of problems, not just one example at a time.
Linear equations in two variables, like
$$
2x + 3y = 6,
$$
play a central role in connecting algebra to the graph of a line in the coordinate plane. You will study how different forms of linear equations (such as “slope–intercept form”) reveal information about the corresponding line.
Systems of Linear Equations
A system of linear equations is a set of two or more linear equations involving the same variables, for example:
$$
\begin{cases}
2x + y = 7 \\
x - y = 1
\end{cases}
$$
In Algebra I you learn techniques (graphical, substitution, elimination) to find all solutions $(x, y)$ that satisfy all equations at once, or to recognize when no such solution exists or when infinitely many exist.
Systems of equations are especially important for modeling situations with more than one unknown, such as problems involving two prices, two speeds, or two unknown quantities that interact.
Polynomials
A polynomial in one variable is an expression made by adding and subtracting terms of the form $ax^n$, where $n$ is a non-negative integer and $a$ is a constant. For example:
- $2x^3 - 5x + 7$
- $x^2 - 4x + 4$
- $7x - 3$ (which is also a linear expression)
Algebra I introduces:
- The structure of polynomial expressions (terms, coefficients, degree).
- Basic operations with polynomials: adding, subtracting, and multiplying.
- Certain common patterns (like squares and products of binomials) that later lead to factoring techniques.
Polynomials are the main “objects” of algebra. Understanding how they are built and how they behave is essential for later topics such as factoring, solving polynomial equations, and graphing polynomial functions.
Factoring and Quadratic Equations
A central theme in Algebra I is rewriting expressions in different but equivalent forms. Factoring is a way of expressing a polynomial as a product of simpler polynomials.
For example, the polynomial
$$
x^2 + 5x + 6
$$
can be written as
$$
(x + 2)(x + 3).
$$
The second form is “factored” and often easier to work with in solving equations.
A quadratic expression in one variable has the general form
$$
ax^2 + bx + c,
$$
with $a \neq 0$. Equations of the form
$$
ax^2 + bx + c = 0
$$
are called quadratic equations.
Algebra I emphasizes:
- Recognizing quadratic expressions.
- Using factoring (when possible) to solve quadratic equations.
- Introducing other solution methods (such as completing the square and the quadratic formula) in later sections.
Quadratic equations are the first step beyond linearity. They lead to parabolic graphs, have up to two real solutions, and are fundamental in many physical and geometric applications.
Algebraic Thinking vs. Arithmetic Thinking
Algebra I asks you to make a shift in how you think about numbers and operations.
In arithmetic, you usually:
- Start with specific numbers.
- Perform a fixed sequence of operations.
- End with a specific number as an answer.
Example: $7 + 5 \times 3 = 22$.
In algebra, you:
- Work with variables representing unknown or general quantities.
- Manipulate expressions according to rules that must work for all numbers allowed.
- Often end with a formula, a simplified expression, or a set of possible values rather than a single number.
Example: Simplifying $3(x + 2)$ to $3x + 6$ works for any number substituted for $x$.
Algebra I trains you to:
- Recognize and use patterns in operations.
- Justify each transformation of an equation (for example, why you can “do the same thing to both sides”).
- Use equations and inequalities to represent conditions or constraints.
This focus on structure and generality is what allows algebra to apply widely, from financial calculations to physics and computer science.
Graphical Perspective in Algebra I
The coordinate plane and the concept of a function are formally introduced elsewhere, but Algebra I is where you begin to actively use graphs as part of algebraic work.
Within Algebra I, you:
- See how linear equations in two variables correspond to straight lines.
- Learn how parameters in an equation (like slope and intercept) affect the appearance and position of the graph.
- Use graphs to visualize solutions: for instance, the point where two lines cross corresponds to the solution of a system of two linear equations.
- Start to recognize non-linear shapes such as parabolas coming from quadratic equations.
This interplay between symbolic (equations), numeric (tables of values), and visual (graphs) viewpoints is a characteristic feature of Algebra I and sets you up for more advanced function analysis in later courses.
Modeling and Word Problems
Algebra I is also your first major encounter with mathematical modeling in a systematic way. In this context, modeling means translating a situation into algebraic language and then interpreting the algebraic solution back in terms of the original situation.
A typical modeling process in Algebra I looks like this:
- Identify unknowns.
Decide what quantities are not known and assign variables to them (for example, let $x$ be the number of hours worked). - Express relationships.
Use the information given to write equations or inequalities relating those variables (for example, “total pay = hourly rate × hours worked”). - Solve algebraically.
Use your algebra techniques (solving linear equations, systems, or quadratics) to find the variable values that satisfy the relationships. - Interpret and check.
Turn the numeric solution back into a statement about the original problem and check that it makes sense (for example, it should not give a negative number of items or hours).
Algebra I problems will cover contexts like:
- Mixtures (for example, combining solutions of different concentrations).
- Motion (relating distance, rate, and time).
- Financial problems (budgeting, simple interest).
- Geometry problems where unknown sides or areas are expressed algebraically.
These problems show that algebra is not just a collection of symbolic tricks; it is a practical language for understanding and organizing information.
Skills You Develop in Algebra I
Across the specific subtopics that follow this chapter, Algebra I helps you build several broad skills:
- Manipulating expressions.
You learn to expand, factor, and simplify expressions accurately and efficiently. - Solving equations and inequalities.
You practice systematic methods that can be applied and adapted in new situations. - Working with functions and graphs.
You gain familiarity with interpreting and producing graphs for linear and quadratic relationships. - Connecting different representations.
You move between word descriptions, tables, equations, and graphs of the same situation. - Checking and reasoning.
You develop habits of verifying your solutions (by substitution or graphing) and explaining why steps are valid.
These skills are essential not only for higher-level mathematics but also for any situation in which you need to reason about changing quantities and their relationships.
How Algebra I Fits into the Bigger Picture
Algebra I serves as a bridge between basic arithmetic and more abstract or advanced topics.
From earlier topics, it builds on:
- Comfort with numbers and arithmetic operations.
- An initial understanding of variables and simple equations.
- Familiarity with plotting points on the coordinate plane.
It prepares you for:
- Algebra II, which deepens the study of functions, introduces more complex polynomial and rational functions, and adds new types of equations.
- Geometry, where algebraic methods are used to analyze geometric figures, especially in coordinate geometry.
- Trigonometry and Precalculus, where more advanced functions and transformations are central.
- Calculus, where functions, limits, derivatives, and integrals rely on algebraic fluency.
- Statistics and Probability, where algebraic expressions and equations describe distributions, expectations, and models.
By the end of Algebra I, you should feel comfortable treating algebraic expressions and equations as familiar objects you can manipulate, interpret, and use to solve meaningful problems.