You will use points on a line to write a general equation of a line.
After completing this tutorial, you will be able to complete the following:
Recall that two quantities are linearly related if there is a constant ratio of change between the two quantities. In many such relationships one quantity can be viewed as dependent on the other. The variables x and y are frequently (but certainly not always) used in these scenarios; x represents the independent variable, and y represents the dependent variable. Specific values of the dependent variable are uniquely determined by values of the independent variable. With this vocabulary, we say that there is a linear relationship between x and y if there is a constant ratio between the change in y and the change in x. The constant ratio of change is called the rate of change. The slope of a line, a measure of the line's steepness, coincides with the rate of change.
Recall the slope formula, which is used to find the slope of a line given two points on the line:
Each line has two distinguished points at which the line crosses the x- and y-axes. These are called the x- and y-intercepts, respectively. The y-intercept can be viewed as an initial value, since it corresponds to x = 0 in the underlying relationship. A line's slope and intercept can be used to write an algebraic equation that describes the line. This equation also describes the underlying linear relationship, so such equations are called linear equations.
We have numerous types of linear equations. The slope-intercept form of a line is an equation of the form where m is the slope of the line and b is the y-intercept. Notice, however, that the slope-intercept form of a line cannot describe a vertical line that has an undefined slope. Another form of a line is called the general form of a line, which has the form where a, b, and c are real numbers. Any line can be written in general form, an aspect that separates it from slope-intercept. Further, in contrast to slope-intercept form, which is unique, each line has infinitely many general forms.
Rearranging, we get a general form of the line:
Notice that if we multiply the equation through by a constant, we will have another general form of the line. For this reason, a general equation of a line is not unique.
|Approximate Time||20 Minutes|
|Pre-requisite Concepts||Students should know the equation of a line in general form.|
|Type of Tutorial||Skills Application|
|Key Vocabulary||equation of a line in general form, horizontal line, intercept form of a line|