Introduction to Algebra: Variables, Expressions

• Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols.
• These symbols represent quantities without fixed values, known as variables.
• Algebra is a broad field; key concepts are reviewed here.

Variables and Expressions

• A variable is a symbol (usually a letter) that represents a quantity that can change.
• An expression is a combination of variables, numbers, and operation symbols (+, -, ×, ÷).
• Example: 3x + 5 is an algebraic expression where 'x' is the variable.

Equations

• An equation is a statement that two expressions are equal.
• Equations contain an equals sign (=).
• Example: 3x + 5 = 14 is an equation.
• Solving an equation means finding the value(s) of the variable(s) that make the equation true.

Basic Operations

• Addition (+): Combining two or more quantities.
• Subtraction (-): Finding the difference between two quantities.
• Multiplication (× or *): Repeated addition of a quantity.
• Division (÷ or /): Splitting a quantity into equal parts.

Order of Operations

• PEMDAS/BODMAS is a mnemonic for the order of operations:
  • Parentheses/Brackets
  • Exponents/Orders
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
• Example: To evaluate 2 + 3 × 4, multiplication is done first: 2 + 12 = 14.

Combining Like Terms

• Like terms are terms that have the same variable raised to the same power.
• Like terms can be combined by adding or subtracting their coefficients.
• Example: 3x + 2x can be simplified to 5x.

Distributive Property

• The distributive property states that a(b + c) = ab + ac.
• It allows you to multiply a single term by two or more terms inside parentheses.
• Example: 2(x + 3) = 2x + 6

Solving Linear Equations

• A linear equation is an equation in which the highest power of the variable is 1.
• Goal: Isolate the variable on one side of the equation.
• Use inverse operations to isolate the variable.
• Steps:
  • Simplify both sides of the equation.
  • Use addition or subtraction to move terms to the correct side.
  • Use multiplication or division to isolate the variable.
• Example:
  • Solve 2x + 3 = 7
  • Subtract 3 from both sides: 2x = 4
  • Divide both sides by 2: x = 2

Solving for One Variable

• Use algebraic manipulation to isolate the desired variable.
• Apply inverse operations to both sides of the equation to maintain equality.

Inequalities

• An inequality is a statement that compares two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to).
• Solving inequalities is similar to solving equations, but with one key difference:
  • Multiplying or dividing both sides by a negative number reverses the inequality sign.

Graphing Linear Equations

• Linear equations can be graphed on a coordinate plane.
• 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.
• To graph:
  • Plot the y-intercept (b).
  • Use the slope (m) to find another point (rise over run).
  • Draw a line through the two points.

Systems of Equations

• A system of equations is a set of two or more equations with the same variables.
• Solving a system of equations means finding the values of the variables that satisfy all equations simultaneously.
• Methods for solving:
  • Graphing: Find the point of intersection of the lines.
  • Substitution: Solve one equation for one variable and substitute that expression into the other equation.
  • Elimination: Add or subtract the equations to eliminate one variable.

Exponents

• An exponent indicates how many times a base number is multiplied by itself.
• Example: In x³, x is the base and 3 is the exponent, meaning x * x * x.
• Rules of Exponents:
  • xᵃ * xᵇ = xᵃ⁺ᵇ (Product of powers)
  • (xᵃ)ᵇ = xᵃᵇ (Power of a power)
  • xᵃ / xᵇ = xᵃ⁻ᵇ (Quotient of powers)
  • x⁰ = 1 (Zero exponent)
  • x⁻ᵃ = 1/xᵃ (Negative exponent)

Polynomials

• A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• Examples: x² + 3x + 2, 4y - 7, 5 (a constant polynomial)
• Types of Polynomials:
  • Monomial: One term (e.g., 5x)
  • Binomial: Two terms (e.g., x + 2)
  • Trinomial: Three terms (e.g., x² + 3x + 1)
• Operations with Polynomials:
  • Addition/Subtraction: Combine like terms.
  • Multiplication: Use the distributive property.

Factoring

• Factoring is the process of breaking down a polynomial into a product of simpler polynomials.
• Common Factoring Techniques:
  • Greatest Common Factor (GCF): Find the largest factor common to all terms.
  • Difference of Squares: a² - b² = (a + b)(a - b)
  • Perfect Square Trinomial: a² + 2ab + b² = (a + b)² and a² - 2ab + b² = (a - b)²
  • Factoring Quadratics: ax² + bx + c = (px + q)(rx + s)

Quadratic Equations

• A quadratic equation is an equation of the form ax² + bx + c = 0, where a ≠ 0.
• Methods for Solving:
  • Factoring: Factor the quadratic expression and set each factor equal to zero.
  • Quadratic Formula: x = (-b ± √(b² - 4ac)) / (2a)
  • Completing the Square: Manipulate the equation to form a perfect square trinomial.

Rational Expressions

• A rational expression is a fraction where the numerator and denominator are polynomials.
• Simplifying Rational Expressions:
  • Factor the numerator and denominator.
  • Cancel out common factors.
• Operations with Rational Expressions:
  • Multiplication: Multiply the numerators and denominators.
  • Division: Multiply by the reciprocal of the second fraction.
  • Addition/Subtraction: Find a common denominator and combine the numerators.

Radicals

• A radical is an expression that involves a root, such as a square root (√), cube root (∛), etc.
• Simplifying Radicals:
  • Factor out perfect squares (or cubes, etc.) from the radicand (the expression under the radical).
• Operations with Radicals:
  • Addition/Subtraction: Combine like radicals (radicals with the same radicand).
  • Multiplication: Multiply the radicands and simplify.
  • Division: Rationalize the denominator (eliminate radicals from the denominator).