Equations of Straight Lines

Fundamentals of Straight Line Equations

• The equation of a straight line in the Cartesian coordinate system traditionally takes the form y=mx+c. Here, ‘m’ stands for the slope, or gradient of the line, while ‘c’ is the y-intercept - the point where the line crosses the y-axis.
• The slope of a line can be determined using two points on the line, (x1,y1) and (x2,y2), employing the formula: m = (y2 - y1) / (x2 - x1).
• A line’s y-intercept is found by setting x to 0 in the line’s equation and solving for y.

Intercepts and Intersection Points

• An x-intercept is the point where the line crosses the x-axis, it is found by setting y to 0 in the equation of the line and solving for x.
• When the equation for two lines are set equal to each other and solved, this yields the coordinates of the intersection point of these two lines.

Characteristics of Parallel and Perpendicular Lines

• If two lines are parallel, then their slopes, or gradients, are equal.
• If two lines are perpendicular, the product of their gradients is -1. Hence, the gradient of a perpendicular line is the negative reciprocal of the original line’s gradient.

Special Forms of the Equation of a Straight Line

• The point-slope form of the equation of a straight line allows you to create the equation from a known slope and a single known point on the line - y-y1 = m(x-x1).
• The two-point form is equally useful. When two points on the line are known, this form can be used - (y - y1) = ((y2 - y1) / (x2 - x1)) * (x - x1).
• You can use the intercept form to create the equation of a straight line from its x and y intercepts - x/a + y/b = 1.

Real-world Application of Straight Line Equations

• The principles of coordinate geometry and the equation of straight lines are not exclusive to mathematical problems. They have myriad applications in real life - from calculating the shortest distance between places, to devising optimal travel routes, managing logistics and even in computer graphics and physics.
• The knowledge of the intersection point of two lines plays a key role in many physics concepts and are heavily used in graphic designing and animation, where the intersection points determine the way images and vectors interact.