Equation of a Straight Line

Introduction

In a cartesian plane, a line is defined as a series of x,y coordinates. The equation of a line, or linear equation, is a formula that describes this series — for any 'x' value you put into the equation you can find the corresponding 'y' value (and for any 'y' value there is a corresponding 'x' value). Linear equations are easily identified by the 'x' and 'y' with no exponents or square roots or anything fancy like that.

The Equation of a Straight Line is typically expressed in one of three forms:

Standard Form

In the Standard Form of the equation of a straight line, the equation is expressed as:

Ax + By = C

where A and B are not both equal to zero and A, B, and C are integers whose greatest common factor is 1. Examples of the standard form include:

1. 2x + y = 6
2. 3x – y = -2
3. -4x + y = 0

Point-Slope Form

The Point-Slope Form of a linear equation uses the x,y coordinates of a point on the line and the slope of the line. (Slope is the 'slantiness' of the line, described as the change in the 'y' values over the change in the 'x' values of any two points on the line.)

The Point-Slope Form is expressed as:

y – y1 = m(x – x1)

where 'm' is the slope of the line and (x1, y1) are the coordinates to a point on the line. Examples of the point-slope form include:

1. y – 4 = -2(x – 1)
2. y – 8 = 3(x – 2)
3. y – 12 = 4(x – 3)

Slope-Intercept Form

The Slope-Intercept Form of the equation of a straight line introduces a new concept, that of the y-intercept. The y-intercept describes the point where the line crosses the y-axis. (At this set of coordinates, the 'y' value is zero, and the 'x' value is the y-intercept.)

The Slope-Intercept Form is expressed as:

y = mx + b

where 'm' is the slope of the line and 'b' is the y-intercept. (That means the point (b,0) is where the line cross the y-axis.) Examples of the slope-intercept form include:

1. y = -2x + 6
2. y = 3x + 2
3. y = 4x (or y = 4x + 0)

Moving Between the Forms

If you have any of the forms of an equation of a straight line, you can use a bit of algebra to find the other forms. In fact, if you examine the examples above you'll find that they describe only three lines — all the 1.)'s are the same line, all the 2.)'s are the same, and all the 3.)'s are the same!