Algebra: Variables, Constants, and Expressions

• Algebra is a branch of mathematics that uses symbols to represent numbers and quantities
• It involves solving equations and inequalities to find the values of unknown variables

Variables

• A variable is a symbol, usually a letter, that represents a quantity that can change or vary
• Common variables include x, y, z, a, b, and c
• Variables can represent unknown values that we want to find or quantities that can take on different values in a given problem

Constants

• A constant is a fixed value that does not change
• Constants are usually numbers, such as 2, -5, 3.14 (pi), or e (Euler's number)
• In an algebraic expression or equation, constants remain the same regardless of the values of the variables

Expressions

• An algebraic expression is a combination of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division
• Examples of algebraic expressions: 3x + 2, y^2 - 5, 2ab + c
• Expressions do not contain an equals sign (=)

Equations

• An equation is a statement that two algebraic expressions are equal
• Equations contain an equals sign (=)
• Examples of equations: 2x + 3 = 7, y - 5 = 10, x^2 + 2x + 1 = 0
• The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true

Inequalities

• An inequality is a statement that compares two algebraic expressions using inequality symbols
• Common inequality symbols:
  • < (less than)

  • (greater than)

  • ≤ (less than or equal to)
  • ≥ (greater than or equal to)
  • ≠ (not equal to)
• Examples of inequalities: x + 2 < 5, 2y - 1 ≥ 3, a ≠ 4
• Solving an inequality involves finding the range of values for the variable(s) that satisfy the inequality

Operations

• Basic operations in algebra:
  • Addition (+)
  • Subtraction (-)
  • Multiplication (× or *)
  • Division (÷ or /)
  • Exponentiation (^)
  • Root extraction (√)
• Order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (PEMDAS)

Terms

• A term is a single number, a variable, or a product of numbers and variables
• Terms are separated by addition or subtraction signs in an algebraic expression
• Examples of terms: 3x, -2y^2, 5, ab, -c/4

Coefficients

• A coefficient is the numerical factor of a term that contains a variable
• It is the number that is multiplied by the variable in a term
• Examples:
  • In the term 3x, the coefficient is 3
  • In the term -2y^2, the coefficient is -2
  • In the term ab, the coefficient is 1 (since ab is the same as 1*ab)

Like Terms

• Like terms are terms that have the same variable(s) raised to the same power(s)
• Like terms can be combined by adding or subtracting their coefficients
• Examples of like terms:
  • 3x and 5x (both have the variable x raised to the power of 1)
  • 2y^2 and -4y^2 (both have the variable y raised to the power of 2)
  • 7ab and -ab (both have the variables a and b raised to the power of 1)
• Unlike terms cannot be combined

Polynomials

• A polynomial is an algebraic expression consisting of one or more terms, where each term contains only non-negative integer exponents
• Examples of polynomials:
  • 5x^2 - 3x + 2 (quadratic polynomial)
  • 2x^3 + x - 7 (cubic polynomial)
  • 4 (constant polynomial)
• Polynomials can be classified based on the number of terms:
  • Monomial: A polynomial with one term (e.g., 3x^2)
  • Binomial: A polynomial with two terms (e.g., 2x + 1)
  • Trinomial: A polynomial with three terms (e.g., x^2 - 5x + 6)

Simplifying Expressions

• Combining like terms: Add or subtract the coefficients of like terms to simplify an expression
• Distributive property: Multiply a term by each term inside parentheses
• Factoring: Expressing a polynomial as a product of simpler polynomials or terms

Solving Equations

• Isolate the variable: Use algebraic operations to isolate the variable on one side of the equation
• Addition and subtraction property of equality: Adding or subtracting the same quantity from both sides of an equation does not change the equality
• Multiplication and division property of equality: Multiplying or dividing both sides of an equation by the same non-zero quantity does not change the equality
• Solving linear equations: Use algebraic operations to isolate the variable and find its value
• Solving quadratic equations: Use factoring, completing the square, or the quadratic formula to find the solutions

Solving Inequalities

• Similar to solving equations, but with some differences
• When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed
• Express the solution as a range of values for the variable

Graphing Linear Equations

• Linear equations can be represented graphically as straight lines on a coordinate plane
• 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
• Use the slope and y-intercept or two points on the line to graph the equation

Systems of Equations

• A system of equations is a set of two or more equations containing the same variables
• The solution to a system of equations is the set of values for the variables that satisfy all equations in the system
• Methods for solving systems of equations:
  • Substitution
  • Elimination
  • Graphing