Elementary algebra forms the foundation of higher mathematics, and for most college programs, it's essential. This is where students are introduced to the language of symbols, variables, and equations, and how to represent and solve problems logically and systematically. You can apply these skills to the real world, especially in science, business, and technology.
What Exactly is Elementary Algebra?
Algebra is a branch of mathematics that studies mathematical operations and relationships using symbols and letters to represent numbers and quantities. It is essential to approach the subject with a positive attitude and seek help when needed, as this can significantly impact learning and success in any course.
It involves the manipulation of variables, solving equations, and graphing equations. Elementary algebra is the introductory course that covers the basics of algebraic concepts and methods. In an elementary algebra class, students will learn how to solve equations, work with variables, simplify expressions, and perform basic operations like addition, subtraction, multiplication, and division. They will also learn about linear equations, quadratic equations, polynomials, and exponents. Entirely new for Elementary Algebra? Watch this this video to learn more.
To make your algebra class easier, you should find the ways you learn best and apply them to your class. For instance, are you a good listener? Then, make sure you listen carefully to everything the teacher says in class.
Elementary algebra isn’t just another math requirement; it’s the language used throughout all higher mathematics. Understanding variables and equations now will make calculus, statistics, and even computer science far easier later on.
If you learn best with visual content, make sure to find a way to convert all the written content (text) into visual content. You can turn a quadratic equation into a visual solution easily, and there are tools online that can help you do it.
However, if you are someone who learns by doing (a kinesthetic learner), then you must solve as many problems as you can. Everyone should practice constantly, but it is the best learning methodology for you. If you're a good reader, then we recommend you read every text and theory you can about math. You'll learn about the terms, operations, exponents, applications, and theory all by reading. Any tutor from Superprof can help you discover your strengths and weaknesses as a learner.
Key Concepts in Elementary Algebra
Elementary algebra is a fundamental branch of mathematics that is usually taught to high school seniors or as part of college-level math courses. Topics covered include linear equations, quadratic equations, systems of equations, polynomials, exponents, and rational expressions. The course provides students with a foundation in mathematical concepts and problem-solving skills that will be useful in future math courses and real-world applications.
Variables and Constants
A variable is an unknown or changeable value (e.g. 
). A constant is fixed (e.g., 
, 
, 
). In expressions like 
, 5 is a coefficient, the constant that scales the variable.
Students have to understand the difference between constants and variables since they determine how equations behave. For instance, in a simple cost model
represents the number of items, while 12 and 5 are constants that affect the total.
In algebra, each symbol, whether a coefficient, variable, or exponent, plays a specific role. Writing equations clearly and respecting mathematical order helps prevent sign errors and simplifies problem-solving.
Expressions and Equations
An expression (for example, 
) can be simplified but not "solved". An equation, on the other hand, sets two expressions equal, and can be solved. Simplify by combining like terms and applying the order of operations:
Example of solving a two-step linear equation:
Always check the result by substituting the solution back into the original equation.
Polynomials and Factoring
A polynomial is a sum of terms of the form 
with non-negative integer exponents.
The degree of the polynomial is the highest exponent in any term. Common operations include addition, subtraction, multiplication, and factoring. Factoring rewrites a polynomial as a product to reveal zeros or simplify equations.
Greatest common factor (GCF):
Trinomial factoring:
Difference of squares:
Check every factoring step by expanding to confirm equivalence.
Inequalities and Absolute Values
Inequalities use symbols such as 
, 
, 
, 
.
When solving, remember to reverse the inequality sign if you multiply or divide both sides by a negative number.
Example:
(Notice the flipped sign in the last step.)
Absolute value represents distance from zero, so 
for any number 
.
Solve 
by splitting into two linear equations:
Thus, 
or 
.
Solving Algebraic Equations
One of the most essential skills in elementary algebra, solving algebraic equations, is about finding the value of unknown variables so that a mathematical statement is true. Whether the equation is linear, quadratic, or part of a larger system, there are logical steps for simplifying, substituting, and balancing both sides. Here are the techniques you should master as they´ll be helpful for later studies in calculus, data analysis, and real-world applications like budgeting and scientific modeling.
Linear Equations
A linear equation has the form 
and generally has one solution when 
. Isolate the variable by using inverse operations:
Check: 
.
Linear relationships are also expressed as 
, where 
is the slope (rate of change) and 
is the intercept.
Quadratic Equations
Quadratic equations take the general form 
.
They can have zero, one, or two real solutions.
Methods:
- Factoring (when possible):
Completing the square: transforms an equation into vertex form to solve or graph.
Quadratic formula: works for every quadratic:
The discriminant 
tells you how many real solutions exist:
→ two real, 
→ one real, 
→ complex.
Systems of Equations
A system of equations is a set of two or more equations that share variables.
Solutions satisfy all equations simultaneously.
Systems of equations appear everywhere, from calculating intersection points in engineering to balancing business budgets or predicting market outcomes. Solving for multiple unknowns is a real-world skill.
Methods
Substitution
Solve one equation for a variable, then substitute.
Elimination
Add or subtract equations to eliminate one variable.
Example (elimination):
Possible outcomes:
One solution
Lines intersect.
No solution
Lines are parallel.
Infinitely many solutions
Equations represent the same line.
| Type of Equation | General Form | Graph Shape | Number of Solutions | Typical Real-World Use |
|---|---|---|---|---|
| Linear Equation | y = mx + b | Straight line | One solution (where a ≠ 0) | Calculating cost, distance, or rate of change |
| Quadratic Equation | ax^2 + bx + c = 0 | Parabola (opens up or down) | Up to two real solutions | Projectile motion, optimization, area problems |
| System of Equations | { 2x + y = 7 ; x - y = 1 } | Intersection of two or more lines | One, none, or infinite | Comparing prices, finding equilibrium points, solving mixtures |
Is Elementary Algebra the Same as Algebra 1?
Elementary algebra or algebra 1 is a comprehensive program that provides students with the necessary support and instruction needed to learn linear equations, inequalities, and functions.
Algebra 1 is a challenging course that covers many topics, such as functions, geometry, and statistics. It involves solving equations using numbers and letters, which can be overwhelming for some students.
How Can I Apply Algebra in Real Life?
Algebra is deemed a complex subject because it has many applications. Some of them are in fields such as science, engineering, finance, and technology. It can be used to solve complex problems and make predictions about real-world situations. Here are some examples of how algebra can be applied in real life:
- Calculating the cost of a shopping trip: Algebra can be used to determine the total price, including taxes and discounts. By using algebraic expressions and equations, you can easily choose the final cost of your purchases.
- Designing computer algorithms: Algebra is used extensively in computer science to create algorithms and solve complex problems. Algorithms are essentially sets of equations that need to be solved to produce a specific output.
- Predicting the outcome of experiments: Algebra is used in science to make predictions. By using mathematical equations, scientists can predict how a system will behave under certain conditions.
- Managing finances: Algebra can be used to manage personal finances, such as calculating interest on loans, determining the monthly payment on a mortgage, and creating a budget.
To learn more about the many applications of this fascinating subject, you should read a chapter or two written about solutions provided by algebra. As a student, you should always be searching for how the things you learn about in class are applicable in the real world. Solving a quadratic equation, polynomials, the square root (sqrt), or finding solutions to integers shouldn't be something you automatically do without questioning how this applies in reality.
In the end, every student in the world is taught these subjects not to pass a test or a final exam in various courses. Every student learns universal subjects, like mathematics (addition, multiplication, equations, integers, rational numbers, and more), to help them understand the world. Learning to pass a test is not learning; it's just passing a class. There are plenty of other variables to learning than getting an A on a test.
Have you been paying attention? Here are some quick questions to test your understanding. Good luck!
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