Defining Linear Equations

Defining Linear Equations

• A linear equation is an equation that can be graphed as a straight line.
• It involves only constants and variables raised to the first power.
• Standard form of a linear equation in two variables is Ax + By = C, where A, B, and C are constants, and x and y are variables.

Key Components of a Linear Equation

• Variables: Represent unknown quantities (e.g., x, y, z).
• Constants: Represent fixed values (e.g., 2, -5, 10).
• Coefficients: Numbers that multiply variables (e.g., 3 in 3x).
• Terms: The parts of an equation separated by + or – signs (e.g., 2x, -5y, 10).

Types of Linear Equations

• One variable: Equations with only one variable (e.g., 2x + 5 = 11).
• Two variables: Equations with two variables (e.g., y = 2x + 1) that represent a straight line when graphed.
• Three variables: Equations with three variables (e.g., 2x + 3y - z = 7). These describe planes in three-dimensional space.

Solving Linear Equations

• Isolate the variable: Use inverse operations (addition, subtraction, multiplication, division) to get the variable by itself on one side of the equation.
• Maintain equality: Performing the same operation on both sides of the equation is crucial to maintain the equality.
• Order of operations: Follow the order of operations (PEMDAS/BODMAS) when simplifying.
• Check solutions: Substitute the solution back into the original equation to verify its correctness.

Graphing Linear Equations

• Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
• Slope: Represents the steepness of the line and is calculated as the change in y over the change in x (rise over run).
• Y-intercept: The point where the line crosses the y-axis.
• X-intercept: The point where the line crosses the x-axis.

Special Cases of Linear Equations

• Horizontal lines: Equations of the form y = b have a slope of zero (e.g., y = 3).
• Vertical lines: Equations of the form x = a have an undefined slope (e.g., x = -2).
• Parallel lines: Lines with the same slope but different y-intercepts.
• Perpendicular lines: Lines that intersect at a 90-degree angle. The product of their slopes is -1.

Systems of Linear Equations

• Solutions: Points where the graphs of two or more linear equations intersect.
• Substitution method: Replacing one variable's expression with its equivalent from another equation.
• Elimination method: Adding/subtracting equations to eliminate a variable.
• No solution: Parallel lines (in a two-variable system).
• Infinite solutions: The same line (in a two-variable system).

Applications of Linear Equations

• Real-world problems: Modeling relationships between variables (e.g., cost and revenue, distance and time).
• Geometry: Finding equations of lines given points or slope and a point.
• Business: Predicting future values based on linear relationships (e.g., simple interest formulas).
• Physics: Describing motion or other physical phenomena with linear relationships.

Important Concepts

• Consistent system: Has at least one solution.
• Inconsistent system: Has no solution.
• Dependent system: Has infinitely many solutions.

Practice Problems (Examples)

• Find the slope of the line passing through (2, 5) and (4, 9).
• Find the equation of the line passing through ( -1, 3) with a slope of 2.
• Solve the system of equations: 2x + y = 5 and x - y = 1.