Linear Equations: Slope-Intercept & Point-Slope Forms

Slope-Intercept Form

• This equation describes a straight line on a Cartesian coordinate system, which is essential in algebra and graphing. The equation is typically expressed in slope-intercept form, denoted as y = mx + b. In this formula:
  • ‘m’, representing the slope of the line, indicates the steepness or angle of the line relative to the horizontal axis. A positive value for ‘m’ means the line ascends from left to right, while a negative value indicates a descent.
  • ‘b’, on the other hand, is the y-intercept, which signifies the specific point where the line intersects the y-axis. This is a crucial point as it provides a starting value for the function when the variable x is equal to zero.
• The slope (m) indicates the rate of change of y with respect to x.
• A positive slope indicates an upward trend from left to right, a negative slope indicates a downward trend.
• A slope of zero indicates a horizontal line.
• An undefined slope indicates a vertical line.
• The y-intercept (b) provides the starting point on the y-axis.
• This form is useful for graphing lines quickly, as you know the slope and the point on the y-axis the line passes through.

Point-Slope Form

• Represents a linear equation in the form y - y₁ = m(x - x₁), where:
  • 'm' is the slope of the line
  • '(x₁, y₁)' is a point on the line.
• Useful when you know the slope and a point on the line, or when given two points.
• Enables you to express the equation of a line through a calculation between two points, or one point and the slope. Essentially, using a known point and knowing the slope.
• The point-slope form allows you to find the equation of a line without needing the y-intercept. This can be more efficient when the y-intercept is not easily determined

Vertical and Horizontal Slopes

• Vertical Lines:

  • Have an undefined slope.
  • The equation of a vertical line is always in the form x = a, where 'a' is the x-coordinate of any point on the line.
  • The 'x' value never changes— it is constant.

• Horizontal Lines:

  • Have a slope of zero (m = 0).
  • The equation of a horizontal line is always in the form y = b, where 'b' is the y-coordinate of any point on the line.
  • The 'y' value never changes— it is constant.

• Relationship between Slope and Equation:

  • The slope dictates the direction and steepness of the line.
  • Understanding the slope and a point on the line are key to determining the equation in various forms.

• Real-world applications:

  • Used in modeling various real-world scenarios involving linear relationships. Examples include calculating the growth of a population or determining the distance traveled at a constant speed.