Algebra Basics Quiz

Algebra

Definitions

• Algebra: A branch of mathematics dealing with symbols and the rules for manipulating those symbols to solve equations and represent relationships.
• Variable: A symbol (often x, y, z) that represents an unknown value.
• Constant: A fixed value that does not change.

Basic Concepts

• Expression: A combination of numbers, variables, and operations (e.g., 2x + 3).
• Equation: A statement that two expressions are equal (e.g., 2x + 3 = 7).
• Inequality: A mathematical statement indicating that one expression is greater than or less than another (e.g., x + 2 > 5).

Operations

• Addition: Combining two or more quantities.
• Subtraction: Finding the difference between quantities.
• Multiplication: Repeated addition of a number.
• Division: Splitting a quantity into equal parts.

Solving Equations

1. Isolate the variable: Use inverse operations to get the variable by itself on one side of the equation.
2. Perform the same operation on both sides to maintain equality.
3. Check your solution by substituting the value back into the original equation.

Types of Equations

• Linear Equations: Equations of the form ax + b = c, where a, b, and c are constants.
• Quadratic Equations: Equations of the form ax² + bx + c = 0, which can be solved using factoring, completing the square, or the quadratic formula.
• Polynomial Equations: Equations involving terms that are polynomials (e.g., ax^n + bx^(n-1) + ... + k = 0).

Factoring

• Factoring: The process of breaking down a polynomial into simpler components (e.g., x² - 9 = (x - 3)(x + 3)).
• Common methods:
  • Greatest Common Factor (GCF)
  • Difference of Squares
  • Trinomials

Functions

• Function: A relation that assigns exactly one output for each input (e.g., f(x) = 2x + 3).
• Domain: The set of possible inputs (x-values) for the function.
• Range: The set of possible outputs (y-values) for the function.

Graphing

• Coordinate Plane: A two-dimensional surface on which points are plotted using pairs of numbers (x, y).
• Slope: The measure of the steepness of a line, calculated as rise/run.
• Intercepts: Points where the graph intersects the axes; x-intercept (when y=0) and y-intercept (when x=0).

Systems of Equations

• System of Linear Equations: A set of two or more linear equations with the same variables.
• Methods to solve:
  • Graphing
  • Substitution
  • Elimination

Inequalities

• Solving Inequalities: Similar to equations but pay attention to the direction of the inequality when multiplying/dividing by a negative number.
• Interval Notation: A way to write the solution set of inequalities (e.g., x < 5 is written as (-∞, 5)).

These notes cover fundamental aspects of algebra, providing a foundation for further study and application in mathematics.

Algebra Definitions

• Algebra involves symbols and rules for manipulating them to solve equations and represent mathematical relationships.
• A variable symbolizes an unknown value, commonly represented by letters like x, y, or z.
• A constant signifies a fixed value that remains unchanged.

Basic Concepts

• An expression combines numbers, variables, and operations, for example, 2x + 3.
• An equation asserts that two expressions are equal, such as 2x + 3 = 7.
• An inequality represents the relationship where one expression is greater than or less than another, e.g., x + 2 > 5.

Operations

• Addition is the process of combining two or more quantities.
• Subtraction involves finding the difference between numbers.
• Multiplication can be understood as repeated addition of a number.
• Division entails splitting a quantity into equal parts.

Solving Equations

• Isolate the variable using inverse operations to position it alone on one side of the equation.
• To maintain equality, perform identical operations on both sides.
• Substitute your solution back into the original equation to verify its accuracy.

Types of Equations

• Linear equations are formatted as ax + b = c, where a, b, and c are constants.
• Quadratic equations take the form ax² + bx + c = 0, solvable through factoring, completing the square, or using the quadratic formula.
• Polynomial equations comprise terms that are polynomials, exemplified by ax^n + bx^(n-1) +...+ k = 0.

Factoring

• Factoring is breaking down a polynomial into simpler components; for example, x² - 9 = (x - 3)(x + 3).
• Common factoring methods include the Greatest Common Factor (GCF), the Difference of Squares, and factoring trinomials.

Functions

• A function defines a relation that produces exactly one output for each specific input, such as f(x) = 2x + 3.
• The domain encompasses all possible input values (x-values) for a function.
• The range includes all possible output values (y-values) for a function.

Graphing

• The coordinate plane is a two-dimensional space for plotting points represented by pairs of numbers (x, y).
• Slope indicates the steepness of a line, calculated as the ratio of rise to run.
• Intercepts are points where a graph crosses the axes; specifically, the x-intercept occurs when y=0, and the y-intercept occurs when x=0.

Systems of Equations

• A system of linear equations is formed by two or more linear equations utilizing the same variables.
• Methods to solve these systems include graphing, substitution, and elimination techniques.

Inequalities

• Solving inequalities is akin to solving equations, but attention must be given to the inequality direction when multiplying or dividing by negative numbers.
• Interval notation presents the solution set of inequalities, for example, x < 5 can be expressed as (-∞, 5).

These notes provide a comprehensive overview of key algebra concepts essential for further mathematical studies.