Review of Linear Equations

Types of Linear Equation Forms

• 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.

• Point-Slope Form:

  • Equation: ( y - y_1 = m(x - x_1) )
  • ( m ) is the slope.
  • ((x_1, y_1)) is a point on the line.

Understanding the Slope

• Definition: Rise over run.
• Positive Slope: Line rises.
• Negative Slope: Line falls.
• Horizontal Line: Slope = 0.
• Vertical Line: Slope is undefined.

Calculating Slope Between Two Points

• Given points ((x_1, y_1)) and ((x_2, y_2))
• Slope ( m = \frac{y_2 - y_1}{x_2 - x_1} )

X and Y Intercepts

• X-Intercept: Point where ( y = 0 ).
  • Example: ((3, 0)) means x-intercept is 3.
• Y-Intercept: Point where ( x = 0 ).
  • Example: ((0, 4)) means y-intercept is 4.

Identifying Intercepts from Points

• Example Points: ((2, 5), (-3, 0), (1, 2), (0, 6))
  • X-Intercept: ((-3, 0))
  • Y-Intercept: ((0, 6))

Parallel and Perpendicular Lines

• Parallel Lines:

  • Same slope.
  • Example: If ( m_1 = 2 ), then ( m_2 = 2 ).

• Perpendicular Lines:

  • Slopes are negative reciprocals.
  • Example: ( m_1 = \frac{3}{4} ), ( m_2 = -\frac{4}{3} ).

Graphing Linear Equations

• Slope-Intercept Form: ( y = mx + b )

  • Identify slope and y-intercept.
  • Plot y-intercept.
  • Use slope to find another point.

• Standard Form: ( ax + by = c )

  • Find x and y intercepts.
  • Plot intercepts and connect.

• Point-Slope Form: ( y - y_1 = m(x - x_1) )

  • Identify a point and slope.
  • Plot the point and use slope to find another point.

Graphing Constants

• Y = Constant: Horizontal line. Slope = 0.
• X = Constant: Vertical line. Slope is undefined.

Practice Problems

• Example equation: ( y = 2x - 3 )
  • Identify slope and y-intercept.
  • Determine the correct graph based on slope and intercept.