Understanding Algebra: Variables, Constants & Equations

• Algebra uses mathematical statements to describe relationships between things
• It uses variables to represent numbers that are unknown or can change

Variables

• Variables are symbols, usually letters, representing unknown or changeable values
• Used in algebraic expressions, equations, and formulas
• In "3x + 5", 'x' exemplifies a variable

Constants

• Constants are values that remain unchanged
• They are numbers within algebraic expressions
• In "3x + 5", '3' and '5' are constants

Expressions

• Algebraic expressions combine variables, constants, and arithmetic operations (+, -, ×, ÷)
• They do not include an equals sign (=)
• "3x + 5", "2y - 7", and "a^2 + b^2" are examples

Equations

• Equations are mathematical statements showing the equality of two expressions
• They include an equals sign (=)
• "3x + 5 = 14", "2y - 7 = 3", and "a^2 + b^2 = c^2" are examples

Coefficients

• A coefficient is a number multiplied by a variable
• In "3x + 5", '3' is the coefficient of 'x'

Terms

• A term is a single number, a variable, or numbers and variables multiplied
• Terms are separated by + or - signs in expressions or equations
• In "3x + 5", '3x' and '5' are terms

Like Terms

• Like terms have the same variable raised to the same power
• They can be combined by adding or subtracting their coefficients
• "3x" and "5x" are like terms, but "3x" and "5x^2" are not

Simplifying Expressions

• Simplifying involves rewriting an expression in a simpler form
• It's done by combining like terms and performing operations
• "3x + 5x - 2" simplifies to "8x - 2"

Evaluating Expressions

• Evaluation involves finding an expression's value
• Substitute specific values for variables and perform operations
• To evaluate "3x + 5" when x = 2, substitute: 3(2) + 5 = 6 + 5 = 11

Solving Equations

• Solving an equation means finding the variable values that make the equation true
• Achieved by isolating the variable using inverse operations

Linear Equations

• Linear equations have a maximum variable power of 1
• Their graph is a straight line
• "2x + 3 = 7" exemplifies a linear equation

Solving Linear Equations

• Isolate the variable using inverse operations
• Add or subtract from both sides to eliminate terms
• Multiply or divide both sides to eliminate coefficients
• To solve "2x + 3 = 7", subtract 3: 2x = 4, then divide by 2: x = 2

Quadratic Equations

• Quadratic equations have a maximum variable power of 2
• Standard form is ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0
• "x^2 - 5x + 6 = 0" is an example

Solving Quadratic Equations

• Factoring: Rewrite as (x - p)(x - q) = 0 and solve each factor for x
• Quadratic Formula: Use x = (-b ± √(b^2 - 4ac)) / (2a) to find solutions
• Completing the Square: Rewrite as (x - h)^2 = k and solve for x

Factoring

• Factoring breaks down an algebraic expression into simpler expressions
• Factoring "x^2 - 5x + 6" results in "(x - 2)(x - 3)"

Distributive Property

• It states a(b + c) = ab + ac
• Used to multiply a single term by a group inside parentheses
• "3(x + 2) = 3x + 6"

Order of Operations

• Rules dictating the sequence of arithmetic operations
• PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)
• To evaluate "2 + 3 × 4", multiply first: 2 + 12 = 14

Inequalities

• Mathematical statements showing unequal relationships between expressions
• Uses symbols like <, >, ≤, and ≥
• "x + 3 < 7" is an example

Solving Inequalities

• Similar to solving equations, but flip the inequality sign when multiplying or dividing by a negative number
• To solve "-2x < 6", divide by -2 (and flip the sign): x > -3

Systems of Equations

• Sets of two or more equations with the same variables
• The solution satisfies all equations in the system

Solving Systems of Equations

• Substitution: Solve one equation for a variable and substitute into another
• Elimination: Add or subtract equations to eliminate a variable
• Graphing: Find intersection points of graphed equations for solutions

Functions

• Functions relate inputs to outputs, with each input having exactly one output
• Represented as f(x), where x is the input and f(x) is the output
• "f(x) = 2x + 3"

Graphing Functions

• Plot points (x, f(x)) on a coordinate plane
• The x-axis is for input values, the y-axis for output values

Exponents

• An exponent indicates how many times a base number is multiplied by itself
• In "x^3", 'x' is the base and '3' is the exponent (x^3 = x × x × x)

Rules of Exponents

• Product of Powers: x^a × x^b = x^(a+b)
• Quotient of Powers: x^a / x^b = x^(a-b)
• Power of a Power: (x^a)^b = x^(a×b)
• Power of a Product: (xy)^a = x^a × y^a
• Power of a Quotient: (x/y)^a = x^a / y^a
• Zero Exponent: x^0 = 1 (if x ≠ 0)
• Negative Exponent: x^(-a) = 1 / x^a

Polynomials

• Expressions with variables and coefficients, using addition, subtraction, multiplication, and non-negative integer exponents
• "3x^2 + 2x - 5" is an example

Operations with Polynomials

• Addition and Subtraction: Combine like terms
• Multiplication: Use the distributive property
• Division: Use long division or synthetic division

Radicals

• Radicals (√) indicate the root of a number
• The nth root of x, when raised to the nth power, equals x
• "√9 = 3" (because 3^2 = 9)

Simplifying Radicals

• Factor the radicand into prime factors
• Look for pairs (or n-tuples) and bring them outside the radical
• "√12 = √(2 × 2 × 3) = 2√3"

Rationalizing the Denominator

• Eliminates radicals from the denominator of a fraction
• Multiply the numerator and denominator by a suitable expression
• To rationalize "1/√2", multiply by "√2/√2": (1 × √2) / (√2 × √2) = √2 / 2

Logarithms

• Logarithms are the inverse of exponentiation
• The logarithm of x to the base b is the exponent needed to raise b to produce x
• Written as log_b(x) = y, meaning b^y = x
• "log_2(8) = 3" (because 2^3 = 8)

Properties of Logarithms

• Product Rule: log_b(mn) = log_b(m) + log_b(n)
• Quotient Rule: log_b(m/n) = log_b(m) - log_b(n)
• Power Rule: log_b(m^p) = p × log_b(m)
• Change of Base Formula: log_b(a) = log_c(a) / log_c(b)