Basic Algebra Concepts and Linear Equations

Basic Algebraic Concepts

• Algebra uses symbols (variables) to represent unknown values and numbers to solve equations and inequalities.
• Variables are typically letters (e.g., x, y, z).
• Equations state that two mathematical expressions are equal. Solving an equation involves finding the values of the variables that make the equation true.
• Inequalities state that one mathematical expression is greater than or less than another. Solving inequalities involves finding the values of the variables that make the inequality true.

Solving 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 the variable.
• To solve a linear equation:
  • Combine like terms.
  • Isolate the variable term on one side of the equation.
  • Isolate the variable by performing the inverse operation.
• Example: Solve 2x + 5 = 11.
  • Subtract 5 from both sides: 2x = 6
  • Divide both sides by 2: x = 3

Solving Systems of Linear Equations

• A system of linear equations consists of two or more linear equations with the same variables.
• The solution to a system of linear equations is the set of values of the variables that satisfy all the equations in the system.
• Methods for solving systems include:
  • Graphing: Graph each equation and find the intersection point.
  • Substitution: Solve one equation for one variable and substitute the expression into the other equation.
  • Elimination: Add or subtract the equations to eliminate a variable.

Exponents and Polynomials

• Exponents represent repeated multiplication. For example, x² = x * x.
• A polynomial is an expression that consists of variables and coefficients, combined using addition, subtraction, and multiplication (but not division by a variable).
  • Common polynomial types include monomials (single term), binomials (two terms), and trinomials (three terms).
• Basic operations for polynomial expressions include addition, subtraction, multiplication, and division.

Factoring Polynomials

• Factoring is the process of writing a polynomial as a product of simpler polynomials.
• Common factoring techniques include:
  • Greatest Common Factor (GCF): Identify the largest common factor of all terms.
  • Grouping: Group terms that have a common factor.
• Factoring techniques for special types of polynomials, like quadratic trinomials, are crucial for solving equations.

Quadratic Equations

• A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
• Methods for solving quadratic equations include:
  • Factoring: Factor the quadratic expression and set each factor equal to zero.
  • Quadratic Formula: Use the quadratic formula, which gives the solutions directly from the coefficients a, b, and c.
  • Completing the square: Manipulate the equation to form a perfect square trinomial.

Rational Expressions

• Rational expressions are expressions that can be written as the quotient of two polynomials.
• Operations with rational expressions include:
  • Simplification: Reduce the expression to its lowest terms.
  • Addition and Subtraction: Find a common denominator and combine the numerators.
  • Multiplication and Division: Multiply or divide the numerators and denominators.

Radicals and Exponents

• Radicals represent roots of numbers. For example, √x represents the square root of x.
• Exponents and radicals are related. For example, √x = x^1/2.
• Working with radicals and exponents involves simplifying expressions and applying rules of exponents and radicals.

Functions

• A function is a relationship between two sets where each input corresponds to exactly one output.
• Functions are often written in the form f(x) = ..., where x represents the input and f(x) represents the output.
• Graphing functions helps visualize the relationship between input and output values.

Graphing Linear Equations

• Graphing a linear equation involves plotting the points that satisfy the equation on a coordinate plane.
• The graph is a straight line, and the slope and y-intercept of the line provide crucial information about the equation.
• The slope-intercept form (y = mx + b) is particularly useful for graphing and determining the properties of a line.