Fundamental Concepts of Algebra

Fundamental Concepts

• Variables: Symbols (usually letters) representing unknown values. Used to express relationships mathematically, e.g., x in the equation 2x + 3 = 7.
• Expressions: Combinations of numbers, variables, and operations (addition, subtraction, multiplication, division, etc.). Examples: 3x + 5, 4y - 2.
• Equations: Statements of equality between two expressions. Include an equal sign (=). Example: 2x + 5 = 11. Solving an equation means finding the value of the variable that makes the statement true.
• Inequalities: Show a relationship where one expression is greater than (>) or less than (<) another expression. Includes symbols like ≥ (greater than or equal to) and ≤ (less than or equal to). Example: x + 2 > 7.

Operations

• Order of Operations (PEMDAS/BODMAS): A set of rules for evaluating expressions, typically remembering the order of operations: Parentheses/Brackets, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).
• Addition and Subtraction: Combining like terms. Example: 3x + 2x = 5x, 5 + 2 = 7. Combining unlike terms involves combining those having the same exponents.
• Multiplication: Combining values using a times symbol (*). Examples: 3 * x = 3x, 4 * 5 = 20. Distributing multiplication over addition requires the entire term in parentheses to be multiplied by everything outside. Example: 3(x + 2) = 3x + 6.
• Division: Splitting a value into equal parts. Examples: x/2, 10 divided by 2 = 5. Using the reciprocal to rewrite a division problem as multiplication.
• Properties of Equality: Rules demonstrating how equations can be transformed to maintain equality. Key properties include addition, subtraction, multiplication, and division properties.

Solving Equations and Inequalities

• Solving Equations: Focuses on isolating the variable on one side of the equation using inverse operations. Example: To solve 2x + 5 = 11, subtract 5 from both sides to get 2x = 6 and then divide both sides by 2 to get x = 3.
• Solving Inequalities: Similar to solving equations but using the order properties of inequalities. For example, when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must change.
• Combining Like Terms: A key step in simplifying expressions and solving equations/inequalities. Example: combine 2x + 3x + 7 to get 5x + 7.

Linear Equations

• Slope-Intercept Form: A linear equation written in the form y = mx + b, where ‘m’ is the slope and ‘b’ is the y-intercept. Slope indicates the rate of change and y-intercept is the point where the line intersects the y-axis.
• Standard Form: A linear equation written in the form Ax + By = C. Useful for finding intercepts and graphing.
• Graphing Linear Equations: Representing linear equations on a coordinate plane (x, y axis) by plotting points, using the y-intercept and slope, or by identifying x and y intercepts directly. The points must fall on a straight line, establishing a linear relationship.

Systems of Linear Equations

• Solving Systems Graphically: Graphing both equations on the same coordinate plane and visually identifying the intersection point (if one exists).
• Solving Systems Algebraically (Substitution and Elimination): Substitution involves substituting one variable from one equation into the other equation. Elimination involves performing operations on the equations to eliminate one variable and solve for the other.

Polynomials

• Defining Polynomials: Expressions made up of variables and coefficients, involving non-negative integer exponents. Polynomials can be monomials (one term), binomials (two terms), trinomials (three terms), and more.
• Adding and Subtracting Polynomials: Combining like terms only of polynomials.
• Multiplying Polynomials: Using the distributive property (FOIL method) with products of two binomials and higher degree polynomials.
• Factoring Polynomials: Breaking down a polynomial into simpler expressions that multiply together to obtain the original polynomial.

Exponents and Radicals

• Exponents: Used to represent repeated multiplication. Rules of exponents govern operations involving exponents (e.g., product rule, power rule, etc).
• Radicals: Representations of roots (square roots, cube roots, etc). Simplifying radicals involves removing perfect squares or higher roots from the expression.

Special Cases

• Quadratic Equations: Equations with a variable raised to the second power. Solving these equations can involve factoring, completing the square, or using the quadratic formula.
• Absolute Value: A measure of distance from zero on the number line, always representing a non-negative value. Solving absolute value equations/inequalities involves considering the different possible cases.
• Complex Numbers: Numbers that include the imaginary unit "i", where i^2 = -1. These numbers arise in many mathematical contexts and can be combined using the usual rules of algebra.
• Word Problems: Applying algebraic concepts and techniques to real-world scenarios. These require carefully identifying unknowns, translating problem statements into algebraic expressions or equations, and solving those expressions/equations.