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1. Linear Equations
Definition: A linear equation is an algebraic equation in which the highest power of the variable is 1. It typically involves one or more variables.
Example: 2x + 3 = 7
Standard Form:
One Variable: ax + b = 0 (where 'a' and 'b' are constants)
Two Variables: ax + by = c
Solving:
Isolate the variable: Use inverse operations (addition/subtraction, multiplication/division) to get the variable alone on one side of the equation.
2. Linear Inequalities
Definition: A linear inequality is similar to a linear equation, but instead of an equal sign (=), it uses inequality symbols:
< (less than)
(greater than)
≤ (less than or equal to)
≥ (greater than or equal to)
Example: 2x + 3 < 7
Solving:
Similar to equations: Use inverse operations, but with one key rule:
If you multiply or divide both sides by a negative number, you must flip the inequality sign.
Graphing:
One variable: Represent the solution set on a number line (open or closed circles depending on whether the inequality is strict or inclusive).
Two variables: Represent the solution set as a shaded region on a coordinate plane.
Key Concepts
Variables: Represent unknown quantities (often represented by letters like x, y, z).
Constants: Fixed values.
Coefficients: The numbers that multiply the variables.
Inverse Operations: Operations that undo each other (e.g., addition and subtraction, multiplication and division).
Solution: The value(s) of the variable(s) that make the equation or inequality true.
Applications
Linear equations and inequalities have numerous real-world applications, such as:
Solving word problems: Translating real-world scenarios into mathematical equations or inequalities.
Modeling real-world situations: Representing relationships between variables in various fields like physics, economics, and engineering.
Making decisions: Analyzing constraints and optimizing outcomes based on inequalities.