Main content
Course: Algebra (all content) > Unit 3
Lesson 5: Intro to slope-intercept formIntro to slope-intercept form
Learn about the slope-intercept form of two-variable linear equations, and how to interpret it to find the slope and y-intercept of their line.
What you should be familiar with before taking this lesson
- You should know what two-variable linear equations are. Specifically, you should know that the graph of such equations is a line. If this is new to you, check out our intro to two-variable equations.
- You should also be familiar with the following properties of linear equations:
-intercept and -intercept and slope.
What you will learn in this lesson
- What is the slope-intercept form of two-variable linear equations
- How to find the slope and the
-intercept of a line from its slope-intercept equation - How to find the equation of a line given its slope and
-intercept
What is slope-intercept form?
Slope-intercept is a specific form of linear equations. It has the following general structure. Drum roll ...
Here, and can be any two real numbers. For example, these are linear equations in slope-intercept form:
On the other hand, these linear equations are not in slope-intercept form:
Slope-intercept is the most prominent form of linear equations. Let's dig deeper to learn why this is so.
The coefficients in slope-intercept form
Besides being neat and simplified, slope-intercept form's advantage is that it gives two main features of the line it represents:
- The slope is
. - The
-coordinate of the -intercept is . In other words, the line's -intercept is at .
For example, the line has a slope of and a -intercept at :
The fact that this form gives the slope and the -intercept is the reason why it is called slope-intercept in the first place!
Check your understanding
Why does this work?
You might be wondering how it is that in slope-intercept form, gives the slope and gives the -intercept.
Can this be some sort of magic? Well, it certainly is not magic. In math, there's always a justification. In this section we'll take a look at this property using the equation as an example.
Why gives the -intercept
At the -intercept, the -value is always zero. So if we want to find the -intercept of , we should substitute and solve for .
We see that at the -intercept, becomes zero, and therefore we are left with .
Why gives the slope
Let's refresh our memories about what slope is exactly. Slope is the ratio of the change in over the change in between any two points on the line.
If we take two points where the change in is exactly unit, then the change in will be equal to the slope itself.
Now let's look at what happens to the -values in the equation as the -values constantly increase by unit.
We see that each time increases by unit, increases by units. This is because determines the multiple of in the calculation of .
As stated above, the change in that corresponds to increasing by unit is equal to the slope of the line. For this reason, the slope is .
Want to join the conversation?
No posts yet.