Chapter Notes: Algebra Foundations
Algebra is a powerful tool that helps us solve real-world problems by using letters and symbols to represent unknown quantities. Before you can master algebraic techniques, you need a strong foundation in the basic building blocks. This chapter introduces the essential concepts that form the basis of all algebraic reasoning: understanding variables, working with expressions, using mathematical properties, and recognizing patterns. These skills will prepare you to solve equations, graph functions, and model situations mathematically throughout this course and beyond.
Variables and Expressions
In arithmetic, you work with specific numbers like 5, 12, or 3.7. In algebra, we expand our thinking to include variables-letters that stand in for numbers we don't know yet or numbers that can change. A variable is a symbol, usually a letter like \( x \), \( y \), or \( n \), that represents an unknown or changing quantity.
Variables let us write general rules and relationships. For example, if you earn $15 per hour at a job, the amount you earn depends on how many hours you work. Instead of calculating separately for 1 hour, 2 hours, and 3 hours, we can use a variable \( h \) to represent hours worked. Then your total earnings would be \( 15h \) dollars.
Understanding Algebraic Expressions
An algebraic expression is a mathematical phrase that combines numbers, variables, and operation symbols (like +, -, ×, ÷). Expressions do not have an equals sign; they represent a value but don't make a statement that two things are equal.
Examples of algebraic expressions:
- \( 3x + 7 \)
- \( 2y - 5 \)
- \( 4a + 3b \)
- \( \frac{x}{2} + 10 \)
Each expression has different parts:
- Terms are the parts of an expression separated by addition or subtraction. In \( 3x + 7 \), there are two terms: \( 3x \) and \( 7 \).
- Coefficients are the numerical parts of terms that include variables. In \( 3x \), the coefficient is 3.
- Constants are terms that contain only numbers, with no variables. In \( 3x + 7 \), the constant is 7.
Example: Identify the terms, coefficients, and constants in the expression \( 5x - 2y + 9 \).
What are the parts of this expression?
Solution:
The expression has three terms separated by addition and subtraction: \( 5x \), \( -2y \), and \( 9 \).
The coefficient of \( x \) is 5.
The coefficient of \( y \) is -2.
The constant term is 9.
This expression has three terms, two coefficients (5 and -2), and one constant (9).
Evaluating Expressions
To evaluate an expression means to find its numerical value by substituting specific numbers for the variables. This process is also called substitution.
When evaluating expressions, follow these steps:
- Replace each variable with the given number, using parentheses around the substituted value.
- Follow the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right).
- Simplify to get a single numerical answer.
Example: Evaluate the expression \( 4x + 3 \) when \( x = 5 \).
What is the value of the expression?
Solution:
Substitute 5 for \( x \): \( 4(5) + 3 \)
Multiply: \( 20 + 3 \)
Add: 23
When \( x = 5 \), the expression equals 23.
Example: Evaluate \( 2a^2 - 3b \) when \( a = 4 \) and \( b = 6 \).
What is the value?
Solution:
Substitute the values: \( 2(4)^2 - 3(6) \)
Evaluate the exponent: \( 2(16) - 3(6) \)
Multiply: \( 32 - 18 \)
Subtract: 14
The expression equals 14.
Properties of Real Numbers
Mathematics has several fundamental properties that describe how numbers behave under different operations. Understanding these properties helps you simplify expressions, solve equations, and justify your work. These properties work for all real numbers-whole numbers, fractions, decimals, positive and negative numbers.
Commutative Property
The commutative property says that you can change the order of numbers when adding or multiplying without changing the result.
Commutative Property of Addition:
\[ a + b = b + a \]For example, \( 5 + 8 = 8 + 5 \). Both equal 13.
Commutative Property of Multiplication:
\[ a \times b = b \times a \]For example, \( 3 \times 7 = 7 \times 3 \). Both equal 21.
Note: Subtraction and division are NOT commutative. For instance, \( 10 - 3 \neq 3 - 10 \), and \( 12 \div 4 \neq 4 \div 12 \).
Associative Property
The associative property says that when adding or multiplying three or more numbers, the way you group them doesn't change the result.
Associative Property of Addition:
\[ (a + b) + c = a + (b + c) \]For example, \( (2 + 3) + 4 = 2 + (3 + 4) \). Both equal 9.
Associative Property of Multiplication:
\[ (a \times b) \times c = a \times (b \times c) \]For example, \( (2 \times 5) \times 3 = 2 \times (5 \times 3) \). Both equal 30.
Distributive Property
The distributive property connects multiplication with addition and subtraction. It states that multiplying a number by a sum is the same as multiplying the number by each term inside the parentheses and then adding the results.
\[ a(b + c) = ab + ac \]This property is extremely useful for simplifying expressions and solving equations.
Example: Use the distributive property to simplify \( 6(x + 4) \).
What is the expanded form?
Solution:
Distribute the 6 to both terms inside the parentheses:
\( 6 \times x + 6 \times 4 \)
Simplify: \( 6x + 24 \)
The expression simplifies to \( 6x + 24 \).
The distributive property also works with subtraction:
\[ a(b - c) = ab - ac \]Example: Simplify \( 3(2y - 5) \).
What is the result?
Solution:
Distribute 3 to both terms: \( 3 \times 2y - 3 \times 5 \)
Multiply: \( 6y - 15 \)
The simplified expression is \( 6y - 15 \).
Identity and Inverse Properties
The identity properties tell us which numbers leave other numbers unchanged under certain operations.
Additive Identity: Adding zero to any number gives that same number.
\[ a + 0 = a \]Multiplicative Identity: Multiplying any number by one gives that same number.
\[ a \times 1 = a \]The inverse properties describe how to get back to the identity element.
Additive Inverse: Every number \( a \) has an opposite \( -a \) such that their sum is zero.
\[ a + (-a) = 0 \]Multiplicative Inverse: Every nonzero number \( a \) has a reciprocal \( \frac{1}{a} \) such that their product is one.
\[ a \times \frac{1}{a} = 1 \]Combining Like Terms
When simplifying algebraic expressions, we often need to combine terms that are similar. Like terms are terms that have exactly the same variable parts raised to the same powers. Only the coefficients can differ.
Examples of like terms:
- \( 3x \) and \( 7x \) are like terms (same variable \( x \))
- \( 5y^2 \) and \( -2y^2 \) are like terms (same variable \( y \) with the same exponent)
- \( 4ab \) and \( 9ab \) are like terms (same variables in the same combination)
Examples of terms that are NOT like terms:
- \( 3x \) and \( 3y \) (different variables)
- \( 5x \) and \( 5x^2 \) (different exponents)
- \( 2ab \) and \( 2bc \) (different variable combinations)
To combine like terms, add or subtract their coefficients while keeping the variable part the same.
Example: Simplify \( 4x + 7x \).
What is the simplified expression?
Solution:
Both terms have the variable \( x \), so they are like terms.
Add the coefficients: \( 4 + 7 = 11 \)
Keep the variable: \( 11x \)
The simplified expression is \( 11x \).
Example: Simplify \( 5a + 3b - 2a + 7b \).
What is the result?
Solution:
Identify like terms: \( 5a \) and \( -2a \) are like terms; \( 3b \) and \( 7b \) are like terms.
Combine the \( a \) terms: \( 5a - 2a = 3a \)
Combine the \( b \) terms: \( 3b + 7b = 10b \)
Write the simplified expression: \( 3a + 10b \)
The simplified form is \( 3a + 10b \).
Order of Operations
When evaluating or simplifying expressions, you must follow a specific sequence of steps called the order of operations. This ensures that everyone gets the same answer when calculating the same expression. The standard order is remembered by the acronym PEMDAS:
- Parentheses (or other grouping symbols like brackets)
- Exponents (including powers and roots)
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
Think of PEMDAS as a hierarchy: operations higher in the list must be performed before operations lower in the list. Multiplication and division have equal priority, so perform them from left to right as they appear. The same applies to addition and subtraction.
Example: Simplify \( 3 + 4 \times 2 \).
What is the value?
Solution:
Multiplication comes before addition in the order of operations.
First multiply: \( 4 \times 2 = 8 \)
Then add: \( 3 + 8 = \) 11
The correct answer is 11, not 14.
Example: Simplify \( 20 \div 4 + 3 \times 2 - 1 \).
What is the result?
Solution:
Perform division and multiplication first, working from left to right: \( 20 \div 4 = 5 \) and \( 3 \times 2 = 6 \)
Rewrite the expression: \( 5 + 6 - 1 \)
Perform addition and subtraction from left to right: \( 5 + 6 = 11 \), then \( 11 - 1 = 10 \)
The final answer is 10.
Translating Words to Algebra
One of the most important skills in algebra is translating real-world situations and verbal phrases into algebraic expressions and equations. This skill allows you to use mathematics to solve practical problems.
Common Phrases and Their Translations
Certain words and phrases indicate specific mathematical operations:
Important note: Pay careful attention to word order, especially with subtraction and division. "5 less than a number" translates to \( x - 5 \), not \( 5 - x \). The phrase structure "less than" reverses the order.
Example: Translate the phrase "the sum of three times a number and 7" into an algebraic expression.
What is the expression?
Solution:
Let \( n \) represent the unknown number.
"Three times a number" means \( 3n \).
"The sum of" indicates addition, so we add 7: \( 3n + 7 \)
The algebraic expression is \( 3n + 7 \).
Example: Write an expression for "8 less than the product of 5 and a number."
What is the expression?
Solution:
Let \( y \) represent the number.
"The product of 5 and a number" is \( 5y \).
"8 less than" means we subtract 8 from that product: \( 5y - 8 \)
The expression is \( 5y - 8 \).
Exponents and Powers
An exponent tells you how many times to multiply a number by itself. In the expression \( x^n \), the base is \( x \) and the exponent (or power) is \( n \).
For example:
- \( 5^3 = 5 \times 5 \times 5 = 125 \)
- \( 2^4 = 2 \times 2 \times 2 \times 2 = 16 \)
- \( x^2 \) means \( x \times x \)
Properties of Exponents
When working with exponents, several key properties help simplify expressions:
Product of Powers Property: When multiplying terms with the same base, add the exponents.
\[ x^a \times x^b = x^{a+b} \]For example, \( x^3 \times x^4 = x^{3+4} = x^7 \)
Power of a Power Property: When raising a power to another power, multiply the exponents.
\[ (x^a)^b = x^{ab} \]For example, \( (x^2)^3 = x^{2 \times 3} = x^6 \)
Power of a Product Property: When raising a product to a power, raise each factor to that power.
\[ (xy)^a = x^a y^a \]For example, \( (3x)^2 = 3^2 \times x^2 = 9x^2 \)
Zero Exponent Property: Any nonzero number raised to the power of zero equals one.
\[ x^0 = 1 \quad \text{(where } x \neq 0\text{)} \]Example: Simplify \( x^5 \times x^3 \).
What is the simplified form?
Solution:
Both terms have the same base \( x \).
Use the product of powers property: add the exponents.
\( x^{5+3} = \) \( x^8 \)
The simplified expression is \( x^8 \).
Patterns and Sequences
Algebra often involves recognizing and describing patterns. A sequence is an ordered list of numbers that follow a specific rule. Each number in a sequence is called a term.
Arithmetic Sequences
An arithmetic sequence is a sequence where each term is found by adding the same number (called the common difference) to the previous term.
For example, the sequence 3, 7, 11, 15, 19, ... is arithmetic because we add 4 to each term to get the next one. The common difference is 4.
The general form for the \( n \)th term of an arithmetic sequence is:
\[ a_n = a_1 + (n-1)d \]where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number.
Example: Find the 10th term of the arithmetic sequence: 5, 9, 13, 17, ...
What is the 10th term?
Solution:
Identify the first term: \( a_1 = 5 \)
Find the common difference: \( d = 9 - 5 = 4 \)
Use the formula with \( n = 10 \): \( a_{10} = 5 + (10-1) \times 4 \)
Calculate: \( a_{10} = 5 + 9 \times 4 = 5 + 36 = \) 41
The 10th term is 41.
Geometric Sequences
A geometric sequence is a sequence where each term is found by multiplying the previous term by the same number (called the common ratio).
For example, the sequence 2, 6, 18, 54, ... is geometric because we multiply each term by 3 to get the next one. The common ratio is 3.
The general form for the \( n \)th term of a geometric sequence is:
\[ a_n = a_1 \times r^{n-1} \]where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.
Example: Find the 6th term of the geometric sequence: 4, 12, 36, 108, ...
What is the 6th term?
Solution:
Identify the first term: \( a_1 = 4 \)
Find the common ratio: \( r = \frac{12}{4} = 3 \)
Use the formula with \( n = 6 \): \( a_6 = 4 \times 3^{6-1} = 4 \times 3^5 \)
Calculate: \( 3^5 = 243 \), so \( a_6 = 4 \times 243 = \) 972
The 6th term is 972.
Introduction to Equations
An equation is a mathematical statement that two expressions are equal. It contains an equals sign. Unlike expressions, equations make a claim that can be true or false.
Examples of equations:
- \( x + 5 = 12 \)
- \( 2y - 3 = 11 \)
- \( 3(a + 4) = 21 \)
A solution to an equation is a value for the variable that makes the equation true. To check if a number is a solution, substitute it into the equation and see if both sides are equal.
Example: Is \( x = 7 \) a solution to the equation \( 2x + 3 = 17 \)?
Does this value satisfy the equation?
Solution:
Substitute 7 for \( x \): \( 2(7) + 3 = 17 \)
Simplify the left side: \( 14 + 3 = 17 \)
This gives us: \( 17 = 17 \), which is true.
Yes, \( x = 7 \) is a solution to the equation.
Understanding the difference between expressions and equations is fundamental. An expression can be simplified or evaluated, but an equation can be solved to find the value(s) of the variable that make it true. This foundation prepares you for the next chapter, where you will learn systematic methods for solving various types of equations.
