Linear Equations Overview

Defining Linear Equations

• A linear equation is an equation that can be graphically represented by a straight line.
• It typically involves one or more variables, and the highest power of each variable is 1.
• Examples include: y = 2x + 1, x - 3y = 5, 2x + 3y - z= 6

Standard Form of a Linear Equation

• A linear equation in two variables, "x" and "y", can be expressed in standard form as Ax + By = C, where A, B, and C are constants, and A and B are not both zero.
• This form is useful for identifying the intercepts (where the line crosses the x and y axes).

Slope-Intercept Form

• Another common form of a linear equation is the slope-intercept form, written as y = mx + b.
• In this form, 'm' represents the slope of the line, and 'b' represents the y-intercept (the point where the line crosses the y-axis).
• The slope indicates the rate of change of y with respect to x.

Finding the Slope

• The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is calculated as m = (y₂ - y₁) / (x₂ - x₁).
• A positive slope indicates that the line rises from left to right; a negative slope indicates that the line falls from left to right.
• A slope of zero indicates a horizontal line.
• An undefined slope means the line is vertical.

Graphing Linear Equations

• To graph a linear equation, identify two points on the line.
• These points can be found by substituting values for one variable (either x or y) and solving for the other.
• Plotting these points and drawing a straight line through them represents the graph of the equation.

Solving Linear Equations

• Linear equations can be solved for one variable.
• The goal is to isolate the variable on one side of the equation through various algebraic operations such as addition, subtraction, multiplication, and division.
• The solution is the value of the variable that makes the equation true.

Systems of Linear Equations

• A system of linear equations involves two or more linear equations.
• The solution is the set of values for the variables that satisfy all the equations in the system.
• Techniques for solving systems of linear equations include graphing, substitution, and elimination.

Applications of Linear Equations

• Linear equations are crucial in various fields such as:
  • Geometry: Finding the equation of a line.
  • Physics: modeling relationships.
  • Business: calculating total cost or revenue.
  • Finance: predicting future profits or losses.
  • Engineering: modeling physical systems.
• Linear equations are valuable for modeling and solving problems involving constant rates of change.

Parallel and Perpendicular Lines

• Parallel lines have the same slope.
• Perpendicular lines have slopes that are negative reciprocals of each other.
• This relationship is key in geometric problems and applications.