Linear Functions and Equations, General Form

0 = Ax + By + C

The formula 0 = Ax + By + C is said to be the 'general form' for the equation of a line. A, B, and C are three real numbers. Once these are given, the values for x and y that make the statement true express a set, or locus, of (x, y) points which form a certain line.

Changing to slope-intercept form

At first it is difficult to imagine the slant and position of the line just by looking at the general form. However, we can change the general form into slope-intercept form and then get a good idea of what we are looking at. Here's the algebra:

0 = Ax + By + C Starting equation
0 - By = Ax + By -By + C Subtract By from each side
-By = Ax + C Left side: 0 - By = -By
Right side: By - By = 0
-By/-B = Ax/-B + C/-B Divide each side by -B
y = (-A/B)x + (-C/B) Left side: -By/-B = y
Right side: Ax/-B = (-A/B)x
Right side: C/-B = (-C/B)
y = mx + b Where: m = (-A/B)
Where: b = (-C/B)

Again, where are the slope and y-intercept?

As the previous algebra has shown, if we start with this general form:

0 = Ax + By + C

Then the slope of the line is:

slope = m = -A/B

And the y-intercept is:

y-intercept = b = -C/B

A simple example

Let's start with this general form:

0 = Ax + By + C


A = 2, B = 3, C = 4


0 = 2x + 3y + 4

slope = m = -A/B = -2/3 = -0.67 (approximately)

y-intercept = b = -C/B = -4/3 = -1.33 (approximately)

Our line in slope-intercept form:

y = mx + b

y = -0.67x + (-1.33) or y = -0.67x - 1.33

The following interactive application

Below is a graph that presents a line as it is given by the general form equation. You can control the example values for A, B, and C by clicking their appropriate '+' and '-' buttons. The current general form equation is shown in the upper left corner of the graph.

This is an EZ Math Movie. Click the 'Show system' checkbox to expose the EZ Math Movie language. You do not need to understand EZ Math Movie to use this application.

General form for the equation of a line: 0 = Ax + By + C

Angle measurement: Degrees Radians

Go on input

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