Lecture Notes: Converting Equations to Slope-Intercept Form and Graphing

Converting Standard Form to Slope-Intercept Form

• Equation in Standard Form: Given an equation such as (2x + y = 3).
• Objective: Convert to slope-intercept form (y = mx + b).
• Steps:
  1. Isolate (y) on one side of the equation.
  2. Move terms involving (x) to the other side (e.g., (2x) becomes (-2x)).
  3. The converted equation: (y = -2x + 3).

Identifying Slope and Y-Intercept

• Slope (m): Coefficient of (x). In (y = -2x + 3), the slope (m = -2).
• Y-Intercept (b): Constant term. Here, (b = 3).

Graphing the Equation

• Plotting Y-Intercept:
  • The point ( (0, 3) ) is plotted on the graph.
• Using Slope for Other Points:
  • Slope (-2) = Rise/Run = Down 2 units, Right 1 unit.
  • From ( (0, 3) ), move to ( (1, 1) ).
  • Draw the line through these points.

Example Problem

• Given Equation: (3x - 4y = 12).
• Convert to Slope-Intercept Form:
  1. Move (3x) to the other side: (-4y = -3x + 12).
  2. Divide by (-4) to isolate (y): (y = \frac{3}{4}x - 3).

Identifying Components

• Slope (m): (\frac{3}{4}).
• Y-Intercept (b): (-3).

Graphing Steps

• Plot the Y-Intercept: (-3) on the y-axis.
• Finding Additional Points Using Slope:
  • From ( (0, -3) ), rise 3 units and run 4 units to ( (4, 0) ).
  • Repeat to find more points if needed.
  • Draw the line through these points.

Summary

• Convert equations from standard form to slope-intercept by isolating (y).
• Identify the slope and y-intercept to facilitate graphing.
• Use the slope-intercept form to plot the graph by marking points and drawing a line.