Lesson 1: Introduction to Basic Algebra
Algebra opens up a world of possibilities for solving problems by using letters and symbols to represent unknown values. In this lesson, we’ll dive into the core of algebra: understanding variables and expressions. Variables – often represented by letters like x or y – stand in for unknown numbers, allowing us to create and manipulate expressions that describe various scenarios, from calculating costs to determining quantities. Once we understand these expressions, we’ll explore basic operations with them—adding, subtracting, multiplying, and dividing—equipping you with the skills to simplify and confidently solve algebraic problems. You can work through a PDF worksheet at the end of the lesson to reinforce your learning. Let’s get started!
What is a Variable?
A variable is a symbol, usually a letter, representing an unknown or changeable value in an expression or equation.
In the equation: \(\style{font-size:10px}{2x+5=15}\) the letter \(\style{font-size:20px}x\) is the variable representing an unknown value we aim to solve for.
In the equation: \(\style{font-size:10px}{3x+2y=12}\) both \(\style{font-size:20px}x\) and \(\style{font-size:20px}y\) are variables, representing unknown values that can change depending on the conditions or values assigned to each other.
Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and mathematical operations (like addition, subtraction, multiplication, or division) that represents a value. Unlike an equation, it doesn’t have an equals sign. For example, \(\style{font-size:10px}{3x+2y-5}\) is an algebraic expression.
In an equation like \(\style{font-size:10px}{3x+2y-5=0}\) we can identify terms and coefficients as follows:
Terms are the individual parts of the expression separated by addition or subtraction. In this equation, the terms are \(\style{font-size:10px}{3x}\), \(\style{font-size:10px}{2y}\) and \(\style{font-size:10px}{-5}\)
Coefficients are the numerical factors that multiply the variables. Here, 3 is the coefficient of \(\style{font-size:20px}x\) and 2 is the coefficient of \(\style{font-size:20px}y\). The constant term, −5, does not have a variable and therefore has no coefficient.
Question:
In the expression \(\style{font-size:10px}{4a-3b+7}\) Identify the variables, terms, and coefficients.
Solution:
Variables: The variables in the expression are \(\style{font-size:20px}a\) and \(\style{font-size:20px}b\)
Terms: The terms are \(\style{font-size:20px}4a\), \(\style{font-size:20px}-3b\) and \(\style{font-size:20p
Coefficients: The coefficient of \(\style{font-size:20px}a\) is 4 and the coefficient of \(\style{font-size:20px}b\) is -3. The term 7 is a constant and does not have a variable, so it doesn’t have a coefficient.
Operations with Algebraic Expressions
In algebra, understanding basic operations (addition, subtraction, multiplication, and division) is essential for working with expressions and simplifying complex problems. Let’s have a look at the rules of each operation:
Addition: Combine terms with the same variables to simplify the expression.
Example 1.1
\(\style{font-size:10px}{2x+3x=5x}\)
Example 1.2
Simplify \(\style{font-size:10px}{5a+3b+2a+4b}\)
Group like terms: \(\style{font-size:10px}{(5a+2a)+(3b+4b)}\)
Simplify each group: \(\style{font-size:10px}{7a+7b}\)
Answer: \(\style{font-size:10px}{5a+3b+2a+4b=7a+7b}\)
Subtraction: Subtract terms with the same variable to simplify the expression.
Example 2.1
\(\style{font-size:10px}{10b-3b=7b}\)
Example 2.2
Simplify \(\style{font-size:10px}{7x+3y-4x-2y.}\)
Group like terms: \(\style{font-size:10px}{(7x-4x)+(3y-2y)}\)
Simplify each group: \(\style{font-size:10px}{3x+y}\)
Answer: \(\style{font-size:10px}{7x+3y-4x-2y=3x+y}\)
Multiplication: Expanding expressions by multiplying terms together.
Example 3.1
Expand: \(\style{font-size:10px}{4x(3x+5)}\)
Distribute \(\style{font-size:10px}{4x}\) to each term inside the brackets: \(\style{font-size:10px}{4x⋅3x+4x⋅5}\)
Simplify each term: \(\style{font-size:10px}{12x^2+20x}\)
Answer: \(\style{font-size:10px}{4x(3x+5)=12x^2+20x}\)
Division: Splitting terms by a constant or another variable when possible.
Example 4.1
Given Expression: $$\style{font-size:10px}{\frac{10x^2+5x}{5x}}$$
We can split the fraction so that each term in the numerator is divided by \(\style{font-size:10px}{5x}\):
$$\style{font-size:10px}{\frac{10x^2}{5x}+\frac{5x}{5x}}$$
Now let’s simplify each part:
- First Term: \(\style{font-size:10px}{\frac{10x^2}{5x}}\)
- Rewrite \(\style{font-size:10px}{x^2}\) as \(\style{font-size:10px}{x⋅x}\)
- Now, \(\style{font-size:10px}{\frac{10x^2}{5x}=\frac{10⋅x⋅x}{5⋅x}}\)
- Cancel one \(\style{font-size:20px}x\) from the numerator and denominator $$\style{font-size:10px}{\frac{10⋅x⋅x}{5x}=2x}$$
- Second Term: \(\style{font-size:10px}{\frac{5x}{5x}}\)
- Here, the entire \(\style{font-size:10px}{5x}\) term in the numerator cancels with the \(\style{font-size:10px}{5x}\) in the denominator: $$\style{font-size:10px}{=1}$$
Putting it all together:
$$\style{font-size:10px}{\frac{10x^2+5x}{5x}=2x+1}$$
Simplifying Expressions
Simplifying algebraic expressions is all about reducing them to their simplest form by following a set of core rules. Combining like terms, applying the distributive property, and carefully handling operations can make complex expressions much easier to work with and understand.
Here’s a helpful list of rules to keep in mind when simplifying algebraic expressions:
1. Combine Like Terms:
Terms with the same variable and exponent can be combined by adding or subtracting their coefficients.
Example: \(\style{font-size:10px}{3x+5x=8x}\)
2. Apply the Distributive Property:
Multiply each term inside the parentheses by the term outside the parentheses.
Example: \(\style{font-size:10px}{2(x+3)=2x+6}\)
3. Use the Order of Operations (PEMDAS):
Simplify expressions following the order: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
Example: Simplify \(\style{font-size:10px}{2+3⋅(4-1)}\) by starting with the parentheses: \(\style{font-size:10px}{=2+3⋅3=2+9=11}\)
4. Simplify Fractions by Cancelling Common Factors:
When terms in the numerator and denominator have a common factor, you can cancel them out.
Example: \(\style{font-size:10px}{\frac{6x}3=2x}\)
5. Remove Parentheses by Expanding or Factoring:
Expand when you need to simplify by distributing, and factor when you can group terms to simplify the expression.
Example (Expanding): \(\style{font-size:10px}{3(x+4)=3⋅x+3⋅4=3x+12}\)
Example (Factoring): \(\style{font-size:10px}{4x+8=4(x+2)}\)
6. Watch for Negative Signs and Apply Carefully:
Distribute negative signs when they appear outside parentheses, and remember that subtracting a term is the same as adding its negative.
Example: \(\style{font-size:10px}{-(3x+2)=-3x-2}\)
7. Simplify Powers and Components:
Use exponent rules to simplify terms with the same base. Here is an overview of the key exponent rules:
Key Exponent Rules:
Product of Powers Rule: When multiplying two expressions with the same base, add the exponents. $$\style{font-size:10px}{a^m⋅a^n=a^{m+n}}$$
Quotient of Powers Rule: When dividing two expressions with the same base, subtract the exponents. $$\style{font-size:10px}{\frac{a^m}{a^n}=a^{m-n}}$$
Power of a Power Rule: When raising an expression with an exponent to another power, multiply the exponents. $$\style{font-size:10px}{\left(a^m\right)^n=a^{m⋅n}}$$
Power of a Product Rule: When raising a product to a power, apply the exponent to each factor in the product. $$\style{font-size:10px}{\left(ab\right)^m=a^m⋅b^m}$$
Power of a Quotient Rule: When raising a quotient to a power, apply the exponent to both the numerator and the denominator. $$\style{font-size:10px}{\left(\frac ab\right)^m=\frac{a^m}{b^m}}$$
Zero Exponent Rule: Any base (except zero) raised to the power of zero is equal to 1. $$\style{font-size:10px}{a^0=1\;as\;long\;as\;a\neq0}$$
Negative Exponent Rule: A negative exponent indicates a reciprocal. Move the base to the denominator to make the exponent positive. $$\style{font-size:10px}{a^{-m}=\frac1{a^m}}$$
In this lesson, you explored foundational algebra concepts, starting with variables and expressions and moving through the basic operations of addition, subtraction, multiplication, and division. You learned key rules for simplifying expressions, like combining like terms, using the distributive property, and applying exponent rules—skills that will help you break down complex expressions.
In the next lesson, you’ll learn to evaluate expressions by substituting values for variables and using the PEMDAS technique and other rules for simplifying expressions, equipping you with the foundation to work with more complex algebraic problems.
Now that you’ve mastered the basics of algebraic expressions, give the worksheet a try to test your understanding and sharpen your skills!