General Equation of line

Based on the sources and our conversation history, the General Form is presented as a comprehensive algebraic representation for any straight line in the rectangular coordinate system.

Here’s a breakdown of the key information about the general equation of a line:

• The Equation The general form of the equation of a straight line is given by Ax + By + C = 0.
• Universality This form is powerful because it can represent any straight line, including those that are vertical. Unlike some other forms (like slope-intercept y = mx + c), the general form can handle vertical lines which have an undefined slope.
• Condition for a Line For the equation Ax + By + C = 0 to represent a line, the coefficients A and B cannot be simultaneously equal to 0. Individually, A can be 0 (resulting in a horizontal line) or B can be 0 (resulting in a vertical line), but they cannot both be zero at the same time.
• Relationship to Other Forms All other forms of linear equations, such as the slope-point form, slope-intercept form, two-point form, and intercept form, can be rearranged into this general form.
• Extracting Geometric Properties The general form allows for the determination of geometric properties of the line.
  • Slope (m): For a non-vertical line (where B ≠ 0), the slope can be found by rearranging the equation into the slope-intercept form (y = mx + c). By solving Ax + By + C = 0 for y, we get By = -Ax - C, which gives y = (-A/B)x - C/B. Thus, the slope is m = -A/B. This was shown using the example 3x - 4y + 12 = 0, where A=3 and B=-4, giving a slope of -3/(-4) = 3/4.
  • Y-intercept: For a non-vertical line (where B ≠ 0), the y-intercept occurs where x = 0. Substituting x = 0 into Ax + By + C = 0 gives By + C = 0, so y = -C/B. The y-intercept is (0, -C/B). In the example 3x - 4y + 12 = 0, C=12 and B=-4, so the y-intercept is -12/(-4) = 3 [Conversation history].
  • X-intercept: For a non-horizontal line (where A ≠ 0), the x-intercept occurs where y = 0. Substituting y = 0 into Ax + By + C = 0 gives Ax + C = 0, so x = -C/A. The x-intercept is (-C/A, 0). In the example 3x - 4y + 12 = 0, C=12 and A=3, so the x-intercept is -12/3 = -4 [Conversation history].
• Special Cases (Vertical and Horizontal Lines):
  • If B = 0 (and A ≠ 0), the equation becomes Ax + C = 0, which simplifies to x = -C/A. This represents a vertical line. Vertical lines have an undefined slope.
  • If A = 0 (and B ≠ 0), the equation becomes By + C = 0, which simplifies to y = -C/B. This represents a horizontal line. Horizontal lines have a slope of 0.
• Usefulness The general form is particularly useful for certain calculations, such as finding the distance of a point from a line or the distance between two parallel lines, where the equations are typically given in this form.

In essence, the general equation Ax + By + C = 0 provides a unified algebraic framework to represent all straight lines, offering flexibility in deriving or converting to other forms and facilitating the calculation of key geometric properties.