General Equation of a Line

Mathematically, we define a line as a two-dimensional figure without any width. If we need to join two points, we can join them with an infinite number of points. These points together form a line.

We can represent every line with an equation. We can represent every line with a universal equation, whether it is a curved line or a straight line. In this article, we will discuss the equation of a straight line.

Equation of a straight line

A straight line is a line which neither bends or twists. Examples of straight lines are rays of sunlight, beams of a candle, a ruler, etc. All these travel straight or exist straight in nature.

The universal equation of a straight line is given by: ax + by = c. We also represent the line as: y = mx + c. Here, m is the slope of the line, c is the intercept and x and y are the coordinates of the points along which the line must be drawn. This is known as the slope form of representing the equation of a line.

We can represent a line in various forms: slope-intercept form, general form, standard form, point-slope form, etc.

If we want to join two straight lines, then we need a line to join them. For example, suppose the coordinates of these points are P (x, y) and Q (n, m). Then we need a line to join these coordinates of points.

Forms of equations of a line

The slope is the angle formed between the line and the particular axis we need to construct the line. In other words, the slope is the inclination between a line with the x or y-axis.

One must note that the slopes of the x and y axes are 0 as they are always straight. Therefore, a line parallel to the x-axis will also have a zero slope, and a line parallel to the y axis will also have a zero slope.

Standard equation of a line

The standard equation of line: ax + by = c

Where a, b and c are real numbers.

Consider an equation of line 4x + 6 = 0. In this equation, a = 4, b = 0 and c = -6. Since we do not have any coefficient of y, therefore, the value of y is zero.

Point-slope form

We can find the equation of a line if we know the slope and coordinates of a point.

The point-slope form of the line is given by: y – y1 = m (x – x1).

Here, m is the slope of line

x, and y are the arbitrary points.

x1 and y1 are the coordinates of the point along which we need to draw the line.

Derivation of point-slope form

Let us consider a line with slope m, and a point with coordinates (x1, y1). We know,

Slope = difference in y coordinates / difference in x coordinates

m = (y – y1) / (x – x1)

Multiply both sides by (x – x1), we have,

m (x – x1) = (y – y1)

We can rewrite the above equation as (y – y1) = m (x – x1), the point-slope form.

Slope-intercept form

The slope-intercept form of a line: y = mx + c

Where m is the slope of the line

c is the intercept along the y axis

We can alternatively rewrite this equation as y = mx – d, where d is the x-intercept.

Equation of a line on a graph

We can represent a straight line on a graph by knowing the coordinates of the line. By using the equation ax + by = c, we can find multiple points of x and y and construct a line as shown: