What Is Algebra?

This video gives an overview of Algebra and introduces the concepts of unknown values and variables. It also explains that multiplication is implicit in Algebra.

For more help, check out our Pre-Algebra Lessons Page

Credit goes to Math Antics.

What Is Algebra?

Algebra is a fundamental branch of mathematics that deals with symbols, variables, and the rules for manipulating them to solve equations and describe relationships between quantities. It is considered one of the pillars of mathematics, alongside arithmetic, geometry, and calculus. Algebraic concepts and techniques are pervasive in various fields such as physics, engineering, economics, computer science, and many others.

At its core, algebra enables us to solve problems involving unknown quantities by representing them with symbols or variables. These variables can stand for any number or quantity, allowing us to work with general statements rather than specific numbers. For example, instead of saying “the sum of 3 and 4 is 7,” algebra allows us to express this relationship more generally as “a + b = c,” where ‘a’, ‘b’, and ‘c’ are variables representing numbers.

Algebraic expressions consist of variables, constants, and mathematical operations such as addition, subtraction, multiplication, division, exponentiation, and roots. These expressions can be combined and manipulated using algebraic rules to simplify or transform them. For instance, terms with the same variables and exponents can be combined using the distributive property or factored out using techniques like factoring.

Equations are central to algebra. An equation is a statement asserting that two expressions are equal. Equations typically involve one or more variables and are solved by finding the values of the variables that satisfy the equality. Solving equations involves applying various algebraic operations to isolate the variable on one side of the equation. The solution to an equation is the set of values for the variables that make the equation true.

In addition to equations, algebra also deals with inequalities, which express relationships between quantities that are not necessarily equal. Inequalities use symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) to compare quantities.

Concepts In Algebra

Algebra encompasses several key concepts and techniques, including:

Solving equations – This involves techniques such as isolating the variable, applying inverse operations, and using properties of equality to find the solution set.

Let’s consider a simple linear equation and go through the process of solving it step by step.

Example: 3x + 5 = 17

Step 1: Isolate the variable. To solve for x, we want to get x alone on one side of the equation. In this case, we’ll start by isolating the term containing
x, which is 3x.

3x+ 5 -5 = 17 – 5

3x = 12

Step 2: Apply inverse operations. Now that we have 3x isolated, we need to get rid of the coefficient 3 attached to x. Since 3x means 3 times x, the inverse operation is division by 3.

1/3 x 3x = 1/3 x 12

x = 4

Step 3: Check the solution. It’s always a good idea to check if the solution we found is correct by substituting the value of x back into the original equation.

3(4) + 5 = 17

12 + 5 = 17

17 = 17

Since the left side of the equation equals the right side after substitution, we have confirmed that x = 4 is indeed the solution to the equation.

So, the solution to the equation 3x + 5 = 17 is x= 4.

Graphing – Algebraic equations and inequalities can be represented graphically on coordinate planes. Graphs provide visual representations of relationships between variables and are essential for understanding functions and their behavior.

Functions – Functions describe how one quantity depends on another. They are often expressed algebraically as equations or formulas relating input variables (independent variables) to output variables (dependent variables). Functions play a crucial role in modeling real-world phenomena and are fundamental to calculus.

Example: f(x) = 2x + 3

In this example, f(x) represents a function of x. The function takes an input value x, performs a specific operation on it, and produces an output value.

Here, the function f(x) takes an input x, multiplies it by 2, and then adds 3 to the result. This function can be interpreted as follows:

For any given value of x, when you plug it into the function, the function will output a value equal to 2x + 3.

For instance, if x = 1, then f(1) = 2(1) + 3 = 5.

If x = 2, then f(2) = 2(2) + 3 = 7.

Similarly, if x = -1, then f(-1) = 2(-1) + 3 = 1.

You can represent this function graphically on a coordinate plane, where the x-axis represents the input values and the y-axis represents the output values. Each point on the graph corresponds to a specific input-output pair.

The graph of f(x)=2x+3 would be a straight line with a slope of 2 and a y-intercept of 3. As x increases, the value of f(x) increases at a rate of 2 for every increase of 1 in x, reflecting the linear relationship described by the function.

Polynomials and polynomial functions – Polynomials are algebraic expressions consisting of variables and coefficients, raised to non-negative integer exponents and combined using addition, subtraction, and multiplication. Polynomial functions are functions defined by polynomial expressions.

Factoring – Factoring involves expressing algebraic expressions as products of simpler expressions. Factoring is useful for simplifying expressions, solving equations, and understanding the behavior of polynomial functions.

Rational expressions and equations – Rational expressions are ratios of polynomials, and rational equations involve expressions containing rational functions. Solving rational equations often involves finding common denominators and simplifying the expressions.

Exponents and logarithms – Exponents represent repeated multiplication, while logarithms are the inverse operations of exponentiation. Exponential and logarithmic functions are essential in modeling growth, decay, and various natural phenomena.

Algebra provides a powerful framework for solving a wide range of mathematical problems and analyzing relationships between quantities. Its concepts and techniques serve as the foundation for higher-level mathematics and have practical applications in numerous fields.

How Do We Use Algebra Everyday?

Have you ever stopped to think about how much algebra influences your everyday activities? While it might seem like a distant memory from your school days, algebra is actually deeply embedded in various aspects of our daily lives, often without us even realizing it. From budgeting and cooking to travel and shopping, algebraic concepts play a significant role in helping us make decisions, solve problems, and navigate the complexities of modern life.

One of the most common ways we use algebra in our daily routines is through budgeting and financial management. Whether you’re balancing your checkbook, calculating interest rates on loans, or determining discounts while shopping, algebraic skills such as percentages, equations, and inequalities come into play. Understanding these concepts helps us make informed decisions about our finances and plan for the future effectively.

Cooking and meal preparation also involve algebraic thinking. Adjusting ingredient quantities based on the number of servings needed requires basic algebraic skills, such as proportionality and scaling recipes up or down. Whether you’re doubling a recipe for a dinner party or halving it for a solo meal, algebra helps ensure that your culinary creations turn out just right.

Home improvement projects often require algebraic calculations as well. From measuring dimensions and calculating areas to estimating material costs and budgeting for expenses, algebraic concepts help us plan and execute projects such as painting, remodeling, and landscaping. Whether you’re building a deck, renovating a kitchen, or sprucing up your garden, algebra is there to lend a hand.

When it comes to travel and commuting, algebra comes into play in various ways. Whether you’re calculating distances, speeds, travel times, or fuel consumption, algebraic concepts help you plan your journeys efficiently. From estimating arrival times using public transportation to optimizing routes for a road trip, algebraic thinking helps you get from point A to point B with ease.

Even in leisure activities such as sports and recreation, algebra plays a role. Analyzing statistics, predicting outcomes, and optimizing strategies in sports often involve algebraic reasoning. Whether you’re keeping track of player statistics, understanding scoring systems, or planning game strategies, algebraic thinking helps you make informed decisions and improve your performance.

Beyond these specific examples, algebra teaches us valuable problem-solving skills, logical reasoning, and critical thinking, which are applicable in countless real-world situations. Whether you’re making decisions, analyzing situations, or finding solutions to challenges, algebra provides a powerful framework for reasoning about quantities and relationships, making it an essential skill for navigating daily life effectively.