Finding the Slope and Y-Intercept from Linear Equations

Key Concepts

• Slope-Intercept Form: The equation is in the form ( y = mx + b ), where:
  • ( m ) is the slope of the line.
  • ( b ) is the y-intercept, the point where the line crosses the y-axis ((0, b)).
• Standard Form: To find the slope and y-intercept, convert the equation from standard form to slope-intercept form.
• Equations with No (x) or (y) Terms: Special considerations for determining slope and intercepts.

Examples

Slope-Intercept Form

• Example 1: Equation ( y = 6x + 1 )
  • Slope ( m = 6 )
  • Y-Intercept ( (0, 1) )
• Example 2: Equation ( y = x - 2 )
  • Coefficient of ( x ) is 1, so slope ( m = 1 )
  • Y-Intercept ( (0, -2) )
• Example 3: Equation ( y = -x - 7 )
  • Coefficient of ( x ) is -1, so slope ( m = -1 )
  • Y-Intercept ( (0, -7) )

No Constant Term

• Example: Equation ( y = \frac{2}{3}x )
  • Slope ( m = \frac{2}{3} )
  • Y-Intercept ( (0, 0) )

No (x) Term

• Example: Equation ( y = 2 )
  • Horizontal line, so slope ( m = 0 )
  • Y-Intercept ( (0, 2) )

No (y) Term

• Example: Equation ( x = -3 )
  • Vertical line, slope is undefined
  • No y-intercept

Converting from Standard Form

• Example: Convert ( 2y + 5x = 6 ) to slope-intercept form.

  • Subtract ( 5x ): ( 2y = -5x + 6 )
  • Divide by 2: ( y = -\frac{5}{2}x + 3 )
  • Slope ( m = -\frac{5}{2} ), Y-Intercept ( (0, 3) )

• Example: Convert ( 4y - x = 8 )

  • Add ( x ): ( 4y = x + 8 )
  • Divide by 4: ( y = \frac{1}{4}x + 2 )
  • Slope ( m = \frac{1}{4} ), Y-Intercept ( (0, 2) )

Additional Learning

• Graphing: Use slope and y-intercept to graph linear equations.
• Video Resources: Check links for further examples and explanations on graphing and line properties.

Summary

• The slope is determined by the coefficient of ( x ) in the slope-intercept form.
• The y-intercept is the constant term in the slope-intercept form.
• For horizontal lines, the slope is zero, while for vertical lines, the slope is undefined.

Consider practicing with different forms of equations and transforming them into slope-intercept form to better understand their slopes and y-intercepts.