# General Equation of a Line: ax + by = c

Explore the graph of the general linear equation in two variables that has the form

ax + by = c

using an applet.

by changing parameters a, b and c. The properties of the line such as slope and x and y intercepts are also explored. The investigation is carried out by changing the coefficients a, b, and c and analyzing their effects on the properties of the graph. This large screen interactive java applet helps you explore the graph of the general linear equation in two variables that has the form

ax + by = c

by changing parameters a, b and c. The properties of the graph such as slope and x and y intercepts are also explored. The investigation is carried out by changing the coefficients a, b, and c and analyzing their effects on the properties of the graph.

## Horizontal and Vertical Lines

Vertical lines have equations of the form x = A , where A is a constant.

Horizontal lines have equations of the form Y = B , where B is a constant.

Examples

• x = 2 this is the equation of a vertical line that passes through all points with x coordinate equal to 2.
• y = 0 this is the equation of a horizontal line that passes through all points with y coordinate equal to 0 (x axis).
• x = 0 this is the equation of a vertical line that passes through all points with x coordinate equal to 0 (y axis).
• y = -3 this is the equation of a horizontal line that passes through all points with y coordinate equal to -3.

### Interactive Tutorial Using Java Applet

Your browser is completely ignoring the <APPLET> tag!

• Click on the button above "click here to start" and maximize the window obtained.
• Use the sliders on the left control panel of the applet and set a = 1, b = 0 and
c = 2. You should have a vertical x = 2 (example above part a) that passes through all points with x coordinates equal to 2.
• Set a = 0, b = 1 and c = 0, you should have a horizontal line y = 0 (example above part b) which is the x axis.
• Set a = 1, b = 0 and c = 0, you should have a vertical line x = 0 (example above part c) which is the y axis.
• Set a = 0, b = 1 and c = -3, you should have a horizontal line y = -3 (example above part d) that passes through all points with y coordinate equal to -3.

## x and y Intercepts of the Graph of a Line

We now explore more general equations of lines with equations

ax + by = c, where a is not equal to zero and b is not equal to zero.

The x intercept is found by setting y = 0 in the above equation and solve for x.

ax + b(0) = c
x = c/a
Hence, the x intercept is at (c/a , 0).

The y intercept is found by setting x = 0 in the above equation and solve for x.

a(0) + by = c
y = c/b
Hence, the y intercept is at (0 , c/b).

Example Find the x and y intercepts of the graph of the equations given below.

• 2x - y = 2
• 4x + 2y = 0

Solution to the Questions in the Above Example

• The x intercept is found by solving 2x = 2, which gives x = 1.
The x intercept is at (1 , 0).
Hence, the y intercept by solving -y = 2, which gives y = -2.
The y intercept is at (0 , -2).
• The x intercept is found by solving - 4x = 0, which gives x = 0.
The x intercept is at (0 , 0).
Hence, the y intercept by solving 2y = 0, which gives y = 0.
The y intercept is at (0 , 0).

### Interactive Tutorial Using Java Applet

• Set parameters a = 2, b = -1 and c = 2 in the applet panel. This will define equation in the example above, part a. Locate the x and y intercepts and compare with the solution above.
• Set parameters a = 4, b = 4 and c = 0. This will define equation in the example above, part b. Locate the x and y intercepts and compare with the solution above.
• Set parameters b = 1 and c = 1. Change parameter a. Does the position of the x intercept change? Does the position of the y intercept change? Explain.
• Set parameters a = 1 and c = 1. Change parameter b. Does the position of the x intercept change? Does the position of the y intercept change? Explain.
• Let us write the equation ax + by = c in slope intercept form.
y = -(a/b)x + c/b
The slope is given by -(a/b). Set a, b and c to some values. Drag the red markers so that they are on the line, read their coordinates and find the slope of the line. Compare the slope found to -(a/b).

More pages related to this topic can be found in this site.

1. Slope Intercept Form Of a Line

2. Equations of Line Through Two Points And Parallel and Perpendicular.

3. Slope of a Line

4. Easy to use calculator to find slope and equation of a line through two points.Two Points Calculator

5. Another calculator to find slope, x and y intercepts given the equation of a line.Find Slope and Intercepts of a Line - Calculator

6. Find a Parallel Line Through a Point: Find a line that is parallel to another line and passes through a point.

7. Find a Perpendicular Line Through a Point: Find a line that is perpendicular to another line and passes through a point.

8. Match Linear Equations to Graphs