Cartesian form of a line

Cartesian Form of a Line

Definition and Formulation

• A line in a Cartesian plane can be defined by an equation of the form y = mx + c.
• The “m” in the equation represents the gradient or slope of the line.
• The “c” in the equation is the y-intercept, which is the point at which the line crosses the y-axis.
• The gradient “m” is calculated by the change in the y-coordinates divided by the change in the x-coordinates between any two points on the line.

Determining the Equation of a Line

• To determine the equation of a line, the gradient and a point the line passes through are needed.
• If the gradient and the coordinates of the point are known, the equation can be written in the format y - y1 = m(x - x1), where m is the gradient, and (x1, y1) are the coordinates of the point on the line.
• The y-intercept can be found by substituting x = 0 into the equation of the line.

Parallel and Perpendicular Lines

• Two lines are parallel if their gradients are equal.
• Two lines are perpendicular if the product of their gradients is -1. This means that the gradient of a line perpendicular to a line with gradient m is -1/m.

Application and Significance

• The Cartesian form of a line is used to find the equation of a line, and also to build understanding of concepts such as gradient, intercept, parallelism and perpendicularity.
• This concept plays a significant role in geometry and graph analysis, and is fundamental to more advanced mathematics.
• Remember: Awareness of these properties will greatly help in solving problems involving lines, equations and graphs.