Slope = -intercept=
We explore the different ways we might write the equation of a line including the slope-intercept form, the point-slope form, and standard form.
Slope-Intercept Form of a Line
Recall the previous example where Yara had $50 in her savings account when the year began, and decided to deposit $20 each week without withdrawing any money. In that example, we model using to represent how many weeks have passed. After weeks, Yara has added dollars. And since she started with $50, she has
in her account after weeks. In this example, there is a constant rate of change of 20 dollars per week, so we call that the slope. We also saw that plotting Yara’s balance over time gives us a straight-line graph.
The graph of Yara’s savings has some things in common with almost every straight-line graph. There is a slope, and there is a place where the line crosses the y -axis. We use the symbol, , for the slope of a line.
The -intercept is a point on the -axis where the line crosses. Since it’s on the -axis,
the -coordinate of this point is 0. It is standard to call the point the -intercept, and
call the number the -coordinate of the -intercept.
One way to write the equation for Yara’s savings was where both and are immediately visible in the equation. Now we are ready to generalize this.
When and have a linear relationship where is the slope and is the -intercept, one
equation for this relationship is
and this equation is called the slope-intercept form of the line. It is called this
because the slope and -intercept are immediately discernible from the numbers in the
equation.
What is the slope and -intercept for the line with the following linear equation?
The number is the -value when . Therefore it is common to refer to as the initial
value or starting value of a linear relationship.
Point-Slope Form of a Line
In the previous section, we learned that a linear equation can be written in slope-intercept form, . This section covers an alternative that is often more useful, especially in Calculus: point-slope form.
Starting in 1990, the population of the United States has been growing by about
2.865 million people per year. Also, back in 1990, the population was 253 million.
Since the rate of growth has been roughly constant, a linear model is appropriate.
Let’s try to write an equation to model this.
We consider using , but we would need to know the -intercept, and nothing in the
background tells us that. We’d need to know the population of the United States in
the year 0, before there even was a United States.
We could do some side work to calculate the -intercept, but let’s try something else. Here are some things we know:
- (a)
- The slope equation is
- (b)
- The slope is
- (c)
- One point on the line is because in 1990, the population was 253 million.
If we use the generic to represent a point somewhere on this line, then the rate of change between and has to be 2.965. So
.
While this is an equation of a line, we might prefer to write the equation without using a fraction. Multiplying both sides by gives us
This is a good place to stop. We have isolated , and three meaningful numbers appear in the equation: the rate of growth, a certain year, and the population in that year. This is a specific example of point-slope form.
When and have a linear relationship where is the slope and is some specific point
that the line passes through, one equation for this relationship is
and this equation is called the point-slope form of the line. It is called this because
the slope and one point on the line are immediately discernible from the numbers in
the equation.
Sometimes, it is helpful to be able to express our equation as . To do this when working with the Point-Slope form of a line, all you have to do is add to both sides of the equation. This will give us the Alternate Point-Slope Form.
When and have a linear relationship where is the slope and is some specific point
that the line passes through, one equation for this relationship is
and this equation is called the (alternate) point-slope form of the line. It is
called this because the slope and one point on the line are immediately discernible
from the numbers in the equation.
Note that some people may call this second form the Point-Slope Form of a line. Both ways of writing this form have the advantage that they can be easily written down if you just know a point on the line and the slope of the line.
Standard Form of a Line
We’ve seen that a linear relationship can be expressed with an equation in Slope-Intercept form or with an equation in Point-Slope form. There is a third form that you can use to write line equations. It’s known as standard form.
Imagine trying to gather donations to pay for a $10,000 medical procedure you cannot afford. Oversimplifying the mathematics a bit, suppose that there were only two types of donors in the world: those who will donate $20 and those who will donate $100.
How many of each, or what combination, do you need to reach the funding goal? As in, if people donate $20 and people donate $100, what numbers could and be? The donors of the first type have collectively donated dollars, and the donors of the second type have collectively donated .
So altogether you’d need
This is an example of a line equation in standard form.
It is always possible to write an equation for a line in the form
where ,, and are three numbers (each of which might be 0, although at least one of
and must be nonzero). This form of a line equation is called standard
form.
In the context of an application, the meaning of , , and depends on that context. This equation is called standard form perhaps because any line can be written this way, even vertical lines (which cannot be written using slope-intercept or point-slope form equations).
Intercept Form of a Line
Intercept form of a line is yet another form used to write line equations. It is useful, because you can immediately pick out the - and -intercepts from the equation.
It is sometimes possible to write an equation for a line in the form
where and are two nonzero numbers. This form of a line equation is called
intercept form.
If the -intercept of a line is located at and the -intercept of a line is located at respectively, and and are nonzero, we can write the line in the form . For example, the line with -intercept and -intercept has the equation .
It is important to note that intercept form can only be used when the intercepts occur at non-origin points, so it cannot be used to represent a line that goes through the origin, and cannot be used to represent vertical or horizontal lines.
Special Lines
While we can write the equation of a line in different forms, it is important to note that we can easily rearrange a line given in one form to another form using algebra.
There are two special types of lines which it is worth mentioning at this point.
