The Straight Line and its Equation

Understanding The Concept

In Rectangular Cartesian Coordinates, a straight line is uniquely defined by its slope and the point through which it passes.
The slope of a straight line is defined as the ratio of the vertical change (change in y-coordinate) to the horizontal change (change in x-coordinate) between any two distinct points on the line.
The equation of a straight line is given by:

y = mx + c

where m is the slope of the line and c is the y-coordinate of the point where the line cuts the y-axis, also known as the y-intercept.

The slope or gradient, m, shows the steepness of the line. If m is positive, the line slopes upward as we read from left to right. If m is negative, the line slopes downward.
The y-intercept, c, tells us the exact point at which the line crosses the y-axis. When x is 0, y is c in the line’s equation.
Each point (x, y) that lies on the line will satisfy this equation.
This interpretable form of a line’s equation is called its slope-intercept form y = mx + c.

For example, to find the equation of a line passing through the point (2, 4) with a slope of 3, we’d substitute these values into the slope-intercept form: y = 3x + c. To find c, we substitute x = 2 and y = 4 to get c = -2. So, our line’s equation is y = 3x - 2.
Two lines with the same slope are parallel, while lines that have slopes that multiply to -1 are perpendicular.

Understanding The Straight Line and its Equation is fundamental in the study of Rectangular Cartesian Coordinates.
Comprehending the meaning of the slope and y-intercept in the equation of a line, as well as the ability to work with these quantities in different contexts, are key skills to develop.