Introduction to Algebra: Variables and Expressions

• Algebra is a branch of mathematics using symbols to represent numbers and quantities.
• It provides a framework for expressing relationships and solving problems involving unknown values.

Variables

• Variables are symbols, typically letters, representing unknown quantities or values.
• They allow the generalization of mathematical statements and relationships.
• In the expression 3x + 5, x is a variable.

Constants

• Constants are fixed values that do not change in an expression or equation.
• They provide specific numerical values to mathematical statements.
• In the expression 3x + 5, 3 and 5 are constants.

Expressions

• Expressions are combinations of variables, constants, and mathematical operations, lacking an equals sign (=).
• Expressions cannot be solved like equations.
• 3x + 5, a - b, and 2y^2 are expressions.

Equations

• Equations are mathematical statements showing the equality between two expressions.
• They contain an equals sign (=) and can be solved to find the value(s) of the variable(s) that satisfy the equation.
• 3x + 5 = 14, a - b = 7, and 2y^2 = 8 are equations.

Coefficients

• Coefficients are numerical or constant quantities multiplying a variable in an algebraic term.
• They indicate how many times the variable is being considered.
• In the term 3x, 3 is the coefficient of x.

Terms

• Terms are the individual components of an expression or equation, separated by addition or subtraction signs.
• They can be constants, variables, or products of constants and variables.
• In the expression 3x + 5, 3x and 5 are separate terms.

Operations

• Operations are mathematical processes that combine or manipulate numbers and variables.
• Common operations: addition (+), subtraction (-), multiplication (*), division (/), exponentiation (^), and taking roots.

Order of Operations

• The order of operations dictates the sequence in which operations should be performed in an expression.
• PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) is a mnemonic for remembering the order of operations.

Simplifying Expressions

• Combining Like Terms: Like terms have the same variable raised to the same power.
• Combining like terms involves adding or subtracting their coefficients; 3x + 2x simplifies to 5x.
• Distributive Property: a(b + c) = ab + ac; it removes parentheses by multiplying each term inside by the factor outside.

Solving Linear Equations

• Linear equations are equations where the highest power of the variable is 1.
• The goal in solving a linear equation is to isolate the variable on one side of the equation.
• Inverse operations are used to undo the operations performed on the variable, maintaining equality on both sides.
• Isolating the Variable: Perform the same operation on both sides of the equation to maintain equality.
• Use addition or subtraction to move constants and multiplication or division to isolate the variable.

Solving Systems of Linear Equations

• A system of linear equations contains two or more linear equations with the same variables.
• The solution is the set of values for the variables that satisfy all equations simultaneously.
• Substitution Method: Solve one equation for one variable and substitute that expression into the other equation.
• Elimination Method: Add or subtract the equations to eliminate one variable.

Polynomials

• Polynomials are expressions consisting of variables and coefficients, involving addition, subtraction, and non-negative integer exponents.
• Monomial: A polynomial with one term (e.g., 3x^2).
• Binomial: A polynomial with two terms (e.g., 2x + 5).
• Trinomial: A polynomial with three terms (e.g., x^2 + 3x - 2).

Factoring Polynomials

• Factoring breaks down a polynomial into simpler expressions (factors) that, when multiplied, yield the original polynomial.
• Greatest Common Factor (GCF): Find the largest factor dividing all terms evenly.
• Difference of Squares: a^2 - b^2 = (a + b)(a - b).
• Perfect Square Trinomials: a^2 + 2ab + b^2 = (a + b)^2 or a^2 - 2ab + b^2 = (a - b)^2.

Quadratic Equations

• Quadratic equations are of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
• Factoring: Factor the quadratic expression and set each factor equal to zero.
• Quadratic Formula: For ax^2 + bx + c = 0, the solutions are given by * x = (-b ± √(b^2 - 4ac)) / (2a)*.

Functions

• A function relates a set of inputs (domain) to a set of possible outputs (range), where each input is related to exactly one output.
• Function Notation: Functions are often denoted by f(x), where x is the input and f(x) is the output.

Graphing Linear Equations

• Linear equations can be graphically represented as straight lines on a coordinate plane.
• Slope-Intercept Form: The equation y = mx + b represents a line with slope m and y-intercept b.
• Slope indicates the steepness and direction of the line.
• The y-intercept is the point where the line crosses the y-axis.

Inequalities

• Inequalities are mathematical statements comparing two expressions using inequality symbols such as <, >, ≤, or ≥.
• Solving Inequalities: Similar to solving equations, but multiplying or dividing by a negative number reverses the inequality sign.

Exponents and Radicals

• Exponents indicate the number of times a base is multiplied by itself.
• Radicals (Roots): The nth root of a number a is a value that, when raised to the nth power, equals a.

Logarithms

• A logarithm is the inverse operation to exponentiation.
• If b^y = x, then log_b(x) = y, where b is the base of the logarithm.
• 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).

Absolute Value

• The absolute value of a number is its distance from zero on the number line.
• Denoted by |x|, where |x| is always non-negative.

Rational Expressions

• Rational expressions can be written as a fraction with polynomials in the numerator and denominator.
• Simplifying Rational Expressions: Factor both the numerator and denominator and cancel out common factors.
• Add/Subtract: Find a common denominator and combine the numerators.
• Multiply: Multiply the numerators and denominators.
• Divide: Multiply by the reciprocal of the divisor.

Complex Numbers

• Complex numbers are in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as i^2 = -1.
• Operations with Complex Numbers: Follow the rules of algebra, remembering that i^2 = -1.