Lecture Notes: Linear Models and Systems of Linear Equations

Overview

• Focus: Review of lines in systems of linear equations
• Main Topics: Cartesian coordinate system, plotting ordered pairs, graphing linear equations
• Pages Covered: Page 17 and 18 of lecture notes

Cartesian Coordinate Plane

• A grid system with two perpendicular axes:
  • X-axis: Horizontal
  • Y-axis: Vertical
• Origin: Intersection point of x and y axes, coordinates (0, 0)
• Movement along axes:
  • Negative numbers: Left (x-axis), Lower (y-axis)
  • Positive numbers: Right (x-axis), Upper (y-axis)
• Coordinate representation:
  • Points are denoted as (x, y)
  • Example: Point P with coordinates (-a, b)

Graphing Linear Equations

• Goal: Graph x - (2/5)y = -1 by plotting points
• Method:
  1. Create a table of points
  2. Substitute values for y to find integer x values (eliminate fractions)

Example

• Equation: x = (2/5)y - 1
• Choosing Values for y:
  • y = 0: Results in x = -1, ordered pair (-1, 0)
  • y = -5: Results in x = -3, ordered pair (-3, -5)
  • y = 5: Results in x = 1, ordered pair (1, 5)
• Plotting Points: Plot the points (-1, 0), (-3, -5), and (1, 5)
• Drawing the Line: Connect points to form the line of the equation

Using Technology

• Transition to technology to verify graph
• Equation Rearranged: y = (5/2)x + (5/2)
• Calculator Steps:
  1. Press "y=" button
  2. Enter equation in y= format
     • Use parentheses for fractions
     • Use "alpha" key for x
  3. Press "Graph" button
  4. Adjust window settings if needed (Zoom > Option 6 for standard 10x10)
• Verification: Ensure plotted points match manually plotted graph
  • Points: (-1, 0), (-3, -5), (1, 5)

Conclusion

• Ensure accuracy by comparing manual and calculator-generated graphs
• Key takeaway: Understanding graphing both manually and using technology