Introduction to Linear Equations

Introduction to Linear Equations in Two Variables

• A linear equation in two variables is an equation that can be written in the form ax + by = c, where 'a', 'b', and 'c' are constants, and 'x' and 'y' are variables.
• These equations represent a straight line on a coordinate plane.
• The solution to a linear equation in two variables is a pair of values (x, y) that satisfy the equation.
• Graphically, the solution represents the point where the line intersects the x and y axes.

Graphing Linear Equations

• To graph a linear equation, you can follow these methods:
  • Plot points: Choose at least two values for 'x' (or 'y'), substitute them into the equation to find the corresponding 'y' (or 'x') values. Plot the resulting points on a coordinate plane and draw a straight line through them.
  • Using intercepts: Find the x-intercept (the point where the line crosses the x-axis) by setting y = 0 and solving for x. Find the y-intercept (the point where the line crosses the y-axis) by setting x = 0 and solving for y. Plot these two intercepts and draw a line through them.
• The graph of a linear equation is a straight line. The slope of the line represents the rate of change between x and y.

Slope of a Line

• The slope (m) of a line passing through points (x₁, y₁) and (x₂, y₂) is given by the formula: m = (y₂ - y₁) / (x₂ - x₁)
• The slope of a line indicates the steepness and direction of the line.
• A positive slope indicates an upward trend, while a negative slope indicates a downward trend.
• A horizontal line has a slope of zero.
• A vertical line has an undefined slope.

Standard Form and Slope-Intercept Form

• Standard Form: ax + by = c
• Slope-Intercept Form: y = mx + b, where 'm' is the slope and 'b' is the y-intercept.
• Converting between standard form and slope-intercept form is a crucial skill, as it allows different representations of the same line.

Special Cases of Linear Equations

• Parallel lines: Parallel lines have the same slope but different y-intercepts.
• Perpendicular lines: Two lines are perpendicular if the product of their slopes is -1. The slope of one line is the negative reciprocal of the other line's slope.
• Horizontal lines: Equations of horizontal lines have the form y = k, where 'k' is a constant. The slope is zero.
• Vertical lines: Equations of vertical lines have the form x = h, where 'h' is a constant. The slope is undefined.

Applications of Linear Equations

• Linear equations are used to model many real-world situations, including:
  • Representing relationships between quantities.
  • Solving problems involving constant rate of change.
  • Showing trends in data.
• Word problems can include topics like distance, speed, and time, cost, cost of goods, and other application-based scenarios.

Solving Systems of Linear Equations

• A system of linear equations consists of two or more linear equations with the same variables.
• Methods for solving include:
  • Graphing: The solution is the point where the lines intersect.
  • Substitution: Isolate a variable from one equation and substitute its expression into the other equation.
  • Elimination: Add or subtract the equations to eliminate one variable.
• Understanding the solution (one, none, or infinitely many solutions) is essential.

This quiz covers the fundamentals of linear equations in two variables, including their standard form and graphical representation. It explains plotting points and utilizing intercepts to graph linear equations effectively.