Review of Linear Equations

Forms of Linear Equations

• Slope-Intercept Form:

  • Equation: ( y = mx + b )
  • ( m ) is the slope
  • ( b ) is the y-intercept

• Standard Form:

  • Equation: ( ax + by = c )
  • ( a, b, c ) are coefficients
  • ( x ) and ( y ) are variables

• Point-Slope Form:

  • Equation: ( y - y_1 = m(x - x_1) )
  • ( m ) is the slope
  • Point: ( (x_1, y_1) )

Understanding Slope

• Definition: Slope = rise/run
• Positive Slope: Line rises as you move from left to right
• Negative Slope: Line falls as you move from left to right
• Special Slopes:
  • Horizontal Line: Slope = 0
  • Vertical Line: Slope is undefined

Calculating Slope Between Two Points

• Formula: ( \frac{y_2 - y_1}{x_2 - x_1} )
• Example: Points (2, 5) and (5, 14)
  • Slope = ( \frac{14 - 5}{5 - 2} = 3 )

X and Y Intercepts

• X-Intercept:

  • Where ( y = 0 )
  • Example: Point (3, 0)

• Y-Intercept:

  • Where ( x = 0 )
  • Example: Point (0, 4)

Parallel and Perpendicular Lines

• Parallel Lines:

  • Same slope
  • Symbol: ( \parallel )

• Perpendicular Lines:

  • Slopes are negative reciprocals
  • Symbol: ( \perp )

Graphing Linear Equations

• Using Slope-Intercept Form:

  • Example: ( y = 2x - 4 )
  • Slope = 2, Y-Intercept = -4
  • Plot the y-intercept, use slope to find next points

• Using Standard Form:

  • Find X and Y intercepts
  • Example: ( 3x - 2y = 6 )
  • X-Intercept: ( (2, 0) )
  • Y-Intercept: ( (0, -3) )

• Using Point-Slope Form:

  • Example: ( y - 3 = 2(x - 2) )
  • Point: (2, 3), Slope: 2

Special Lines

• Horizontal Line:

  • Equation: ( y = c )
  • Slope = 0

• Vertical Line:

  • Equation: ( x = c )
  • Slope is undefined

Practice Problems

• Identify the slope and intercepts from given equations and use this to choose the correct graph representation.
• Example: Which graph corresponds to ( y = 2x - 3 )?
  • Slope = 2, Y-Intercept = -3
  • Correct graph has an increasing slope and passes through (0, -3).