Algebra Equations

Equations in Algebra

Definition

• An equation is a statement that says two mathematical expressions are equal.
• It is a balance scale with the same value on both sides.

Types of Equations

• Simple Equations: Equations that can be solved in a single step.
  • Example: 2x = 6
• Linear Equations: Equations of the form ax + by = c, where a, b, and c are constants.
  • Example: 2x + 3 = 7
• Quadratic Equations: Equations of the form ax^2 + bx + c = 0, where a, b, and c are constants.
  • Example: x^2 + 4x + 4 = 0

Properties of Equations

• Addition Property: The same value can be added to both sides of an equation without changing its solution.
• Subtraction Property: The same value can be subtracted from both sides of an equation without changing its solution.
• Multiplication Property: Both sides of an equation can be multiplied by the same non-zero value without changing its solution.
• Division Property: Both sides of an equation can be divided by the same non-zero value without changing its solution.

Solving Equations

• Addition and Subtraction Methods: Isolate the variable by adding or subtracting the same value to both sides of the equation.
• Multiplication and Division Methods: Isolate the variable by multiplying or dividing both sides of the equation by the same non-zero value.
• Inverse Operations: Use inverse operations to isolate the variable.
  • Example: 2x = 6 => x = 6/2 => x = 3

Equations in Algebra

Definition of Equations

• An equation is a statement of equality between two mathematical expressions, similar to a balance scale with the same value on both sides.

Types of Equations

• Simple Equations: One-step solvable equations, e.g., 2x = 6.
• Linear Equations: Equations of the form ax + by = c, where a, b, and c are constants, e.g., 2x + 3 = 7.
• Quadratic Equations: Equations of the form ax^2 + bx + c = 0, where a, b, and c are constants, e.g., x^2 + 4x + 4 = 0.

Properties of Equations

• Addition Property: Adding the same value to both sides of an equation does not change its solution.
• Subtraction Property: Subtracting the same value from both sides of an equation does not change its solution.
• Multiplication Property: Multiplying both sides of an equation by the same non-zero value does not change its solution.
• Division Property: Dividing both sides of an equation by the same non-zero value does not change its solution.

Solving Equations

• Addition and Subtraction Methods: Isolate the variable by adding or subtracting the same value to/from both sides of the equation.
• Multiplication and Division Methods: Isolate the variable by multiplying or dividing both sides of the equation by the same non-zero value.
• Inverse Operations: Use inverse operations to isolate the variable, e.g., 2x = 6 => x = 6/2 => x = 3.

What is Algebra?

• Algebra is a branch of mathematics that deals with the study of variables and their relationships, involving the use of symbols, equations, and formulas to solve problems and model real-world situations.

Key Concepts

• Variables represent unknown values or quantities and are often represented by letters or symbols.
• Constants are numbers that do not change value.
• Algebraic expressions are combinations of variables, constants, and mathematical operations.
• Equations are statements that express the equality of two algebraic expressions.
• Inequalities are statements that express the relative size of two algebraic expressions.

Types of Algebra

• Elementary Algebra covers basic concepts, including linear equations and quadratic equations.
• Intermediate Algebra builds on elementary algebra, covering topics such as systems of equations and functions.
• College Algebra covers advanced topics, including polynomial and rational functions, and series and sequences.

Operations in Algebra

• Addition involves combining two or more algebraic expressions by adding their like terms.
• Subtraction involves combining two or more algebraic expressions by subtracting their like terms.
• Multiplication involves combining two or more algebraic expressions by multiplying their terms.
• Division involves dividing one algebraic expression by another.

Solving Equations

• Linear Equations have the highest power of the variable as 1.
• Quadratic Equations have the highest power of the variable as 2.
• Systems of Equations involve sets of two or more equations that must be solved simultaneously.

Graphing and Functions

• Graphing involves visual representation of algebraic equations on a coordinate plane.
• Functions represent relations between variables, often represented as f(x) = output.
• Domain and Range refer to the set of input values (domain) and output values (range) of a function.