Basic Algebra Concepts and Equations

Basic Algebraic Concepts

• Algebra is a branch of mathematics that uses symbols to represent numbers and relationships between them.
• Variables: Symbols (like x, y, or z) that represent unknown values.
• Constants: Symbols that represent fixed values.
• Expressions: Combinations of variables, constants, and operations (like addition, subtraction, multiplication, division).
• Equations: Statements that show that two expressions are equal. They contain an equals sign.

Solving Basic Equations

• The goal in solving an equation is to isolate the variable on one side of the equation.
• To isolate a variable, perform inverse operations on both sides of the equation (e.g., addition and subtraction, multiplication and division).
• Order of Operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (left to right), Addition and Subtraction (left to right). This order is crucial for evaluating expressions correctly.
• Simplifying expressions is very important before solving equations.

Linear Equations

• A linear equation is an equation that can be written in the form ax + b = 0, where a and b are constants, and x is a variable.
• The graph of a linear equation is a straight line.
• Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
• Point-slope form: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line.

Systems of Linear Equations

• A system of linear equations is a set of two or more linear equations with the same variables.
• Solutions to systems of linear equations can be found using various methods (e.g., substitution, elimination, graphing).
• A system can have one solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (coincident lines).

Polynomials

• Polynomials are algebraic expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Terms in a polynomial are separated by plus or minus signs.
• Degree of a polynomial: The highest power of the variable in any term in a polynomial.

Factoring Polynomials

• Factoring a polynomial means expressing it as a product of simpler polynomials.
• Common Factoring: Identify and factor out common factors from all terms.
• Grouping: Group terms in the polynomial and factor out common terms in each group.
• Difference of Squares: Factoring expressions in the form (x² - y²).
• Quadratic Trinomials: Factoring quadratic expressions in the form ax² + bx + c.

Exponents and Radicals

• Exponents represent repeated multiplication (e.g., x³ = x · x · x).
• Negative exponents mean reciprocals.
• Zero exponents always equal 1.
• Radicals represent roots of numbers (e.g., √x represents the square root of x).
• Laws of exponents and radicals are crucial for calculations.

Quadratic Equations

• Quadratic equations are equations in the form ax² + bx + c = 0, where a, b, and c are constants.
• Solving quadratic equations using factoring, completing the square, and the quadratic formula.
• The quadratic formula: x = (-b ± √(b² - 4ac)) / 2a

Functions

• A function is a special type of relation where each input has only one output.
• Domain: The set of all possible input values (x-values).
• Range: The set of all possible output values (y-values).
• Function notation: f(x) represents the output of the function for a given input x.
• Linear, quadratic, and polynomial functions are part of this concept.