Introduction to Algebra: Expressions and Equations

• Algebra is a branch of mathematics that uses symbols and rules to manipulate them.
• Symbols in algebra represent quantities, known as variables, that do not have fixed values.
• Algebra is applied to solve mathematical problems.
• It deals with equations containing unknowns and inequalities.

Expressions

• Algebraic expressions combine variables, numbers, and algebraic operations.
• Examples of algebraic expressions include: 3x + 2y - 5 and a^2 + b^2.
• Terms are components of algebraic expressions separated by addition or subtraction.
• Constants are terms without variables.
• Coefficients are the numerical parts of terms that include variables.

Equations

• An equation states that two expressions are equivalent.
• Equations contain an equals sign (=).
• Solving an equation means determining the value(s) of the variable(s) that satisfy the equation.
• Equations range from simple to complex, involving single or multiple variables.
• A solution to an equation is a value that, when substituted for the variable, makes the equation true.
• Linear equations have a highest variable power of 1.
• Quadratic equations have a highest variable power of 2.

Basic Operations

• Addition, subtraction, multiplication, and division are fundamental algebraic operations.
• These operations, as in basic arithmetic, are applied to variables and expressions.
• Combining like terms involves adding or subtracting terms with the same variable raised to the same power.
• The distributive property, a(b + c) = ab + ac, multiplies a single term by a group inside parentheses.

Variables and Constants

• Variables are symbols, usually letters, representing unknown or changing quantities.
• Constants are fixed values that remain unchanged in an expression or equation.
• Variables are used to represent numbers.
• Letters commonly denote a variable.

Exponents and Polynomials

• Exponents indicate how many times a base is multiplied by itself.
• x^n signifies x multiplied by itself n times.
• Polynomials, which consist of variables and coefficients, use addition, subtraction, and non-negative integer exponents.
• Polynomial examples include x^2 + 3x - 4 and 5y^3 - 2y + 1.
• A monomial is a polynomial with one term.
• A binomial is a polynomial with two terms.
• A trinomial is a polynomial with three terms.

Factoring

• Factoring breaks down an expression into a product of simpler expressions.
• It reverses the process of expansion
• Factoring solves equations or simplifies algebraic expressions.
• Common factoring techniques:
  • Taking out the greatest common factor (GCF)
  • Difference of squares: a^2 - b^2 = (a + b)(a - b)
  • Perfect square trinomials: a^2 + 2ab + b^2 = (a + b)^2
  • Quadratic trinomials: factoring expressions of the form ax^2 + bx + c

Solving Linear Equations

• Linear equations have the form ax + b = c, with x as the variable, and a, b, and c as constants.
• Steps to solve linear equations:
  • Simplify each side by combining like terms and using the distributive property
  • Isolate the variable term on one side by adding or subtracting constants from both sides.
  • Solve for the variable by dividing both sides by the coefficient of the variable.
• Always verify the solution by substituting it back into the original equation.

Solving Systems of Equations

• A system of equations is a set of two or more equations sharing the same variables.
• Solving a system means finding variable values that satisfy all equations simultaneously.
• Methods for solving systems of equations:
  • Substitution: solve one equation for one variable and substitute that expression into the other equation.
  • Elimination: add or subtract equations to eliminate one variable.
  • Graphing: graph each equation and find the point(s) of intersection.

Inequalities

• An inequality compares two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).
• Solving an inequality means finding the set of values for the variable that satisfy the inequality.
• Solving inequalities is similar to solving equations, but there are key differences:
  • Multiplying or dividing by a negative number reverses the inequality sign.
  • The solution is often a range of values rather than a single value.

Quadratic Equations

• A quadratic equation can be written as ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
• Methods for solving quadratic equations:
  • Factoring: factor the quadratic expression and set each factor equal to zero.
  • Completing the square: manipulate the equation to create a perfect square trinomial on one side.
  • Quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)
• The discriminant (b^2 - 4ac) determines the nature of the solutions:
  • If b^2 - 4ac > 0, there are two distinct real solutions.
  • If b^2 - 4ac = 0, there is one real solution (a repeated root).
  • If b^2 - 4ac < 0, there are no real solutions (two complex solutions).

Functions

• A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
• The input is called the argument of the function, and the output is called the value of the function.
• Functions are commonly denoted by letters such as f, g, or h.
• f(x) represents the value of the function f at the input x.
• Domain: the set of all possible input values for a function.
• Range: the set of all possible output values for a function.
• Types of functions:
  • Linear functions: f(x) = mx + b
  • Quadratic functions: f(x) = ax^2 + bx + c
  • Exponential functions: f(x) = a^x
  • Logarithmic functions: f(x) = log_a(x)
  • Trigonometric functions: f(x) = sin(x), f(x) = cos(x), f(x) = tan(x)

Graphing

• Graphing is a visual representation of algebraic relationships.
• Coordinate plane: a plane formed by two perpendicular number lines, called the x-axis and y-axis.
• Ordered pair: a pair of numbers (x, y) that represents a point on the coordinate plane.
• The graph of an equation is the set of all points (x, y) that satisfy the equation.
• 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.
• Intercepts: the points where a graph intersects the x-axis (x-intercept) or the y-axis (y-intercept).

Explore the basics of algebra, including algebraic expressions and equations. Learn how variables, terms, and constants combine in expressions. Understand equations and how to solve various types of algebraic problems.