The rectangular coordinate system is a powerful tool for visualizing mathematical relationships. It allows us to plot points, draw lines, and represent equations on a two-dimensional plane. This system forms the foundation for graphing and analyzing various mathematical functions.

Understanding the coordinate system is crucial for solving real-world problems. It helps us map locations, analyze data trends, and model relationships between different variables. Mastering this concept opens doors to more advanced mathematical topics and practical applications.

Rectangular Coordinate System

Points on coordinate systems

• Two-dimensional plane formed by intersection of horizontal x-axis and vertical y-axis (also known as the rectangular plane)
  • Origin is point where x-axis and y-axis intersect with coordinates (0, 0)
• Points represented by ordered pairs (x, y)
  • x-coordinate represents horizontal distance from origin
  • y-coordinate represents vertical distance from origin
• Plotting a point involves starting at origin, moving horizontally along x-axis according to x-coordinate, then moving vertically along y-axis according to y-coordinate
• Coordinate plane divided into four quadrants
  • Quadrant I: x > 0 and y > 0
  • Quadrant II: x < 0 and y > 0
  • Quadrant III: x < 0 and y < 0
  • Quadrant IV: x > 0 and y < 0

Tables for linear equations

• Linear equation in two variables $y = mx + b$ represents straight line on coordinate plane
  • m is slope indicating steepness and direction
  • b is y-intercept where line crosses y-axis
• Table of solutions lists pairs of x and y values satisfying equation
  • Create table by choosing several x-values and calculating corresponding y-values using equation
  • Each pair of x and y values represents point on line

Solutions using graphs

• Graph of equation in two variables is set of all points (x, y) satisfying equation
• Verifying if point is solution involves substituting x and y values into equation and checking if equation holds true
• Finding solutions using graph
  1. Plot graph of equation on coordinate plane
  2. Any point lying on graph is solution to equation
  3. x and y coordinates of point represent solution pair

X-coordinates vs y-coordinates

• x-coordinate (also called abscissa) represents horizontal position of point, y-coordinate (also called ordinate) represents vertical position
• Ordered pair (x, y) always written with x-coordinate first followed by y-coordinate
• On graph, x-coordinates increase from left to right, y-coordinates increase from bottom to top

Ordered pairs in real-world contexts

• Ordered pairs represent locations or positions (city grid intersection 3 blocks east and 4 blocks north of origin represented by (3, 4))
• Ordered pairs represent relationship between two quantities (company's sales data with (2, 5) representing 2 years since founding and $5 million in total sales)

Coordinate Geometry

• Cartesian coordinates are used to describe points in the coordinate system
• Coordinate geometry applies algebraic methods to solve geometric problems on the coordinate plane