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The point-slope form of a line's equation is \( y - y_{1} = m(x - x_{1}) \). In this form, \( m \) represents the slope of the line, and \( (x_{1},\ y_{1}) \) are the coordinates of an ordered pair that lies on the line. The variables \( x \) and \( y \) are the independent and dependent variables, respectively.
You may have noticed that the point-slope form of a line's equation is similar to the slope-intercept form of a line. Both of these forms are shown in the table.
| Point-Slope Form |
Slope-Intercept Form |
|---|---|
\( \color{#A80000}{y} - y_{1} = \color{#0A5200}{m}(\color{#0000E0}{x} - x_{1}) \) |
\( \color{#A80000}{y} = \color{#0A5200}{m}\color{#0000E0}{x} + b \) |
Both equations have a slope that is represented by the variable \( \color{#0A5200}{m} \). Both equations also feature an independent variable, \( \color{#0000E0}{x} \), and a dependent variable, \( \color{#A80000}{y} \).
With these similarities, it is not surprising to learn that you can rewrite any equation given in point-slope form into slope-intercept form. In the video below, the instructor will show you how to carry out this process. Pay close attention how the instructor uses the distributive property.
You may want to use the study guide to follow along. If so, click below to download the study guide.
Both point-slope and slope-intercept forms tell us a lot of information about the graph of that equation. Any equation written in point-slope form can also be written in slope-intercept-form, and vice versa. They are just two different ways of expressing a linear equation.
When you have an equation written in point-slope form, and you want to rewrite it in slope-intercept form, the steps to do that are, first, simplify any double negatives. Then distribute on the right-hand side of the equation. Next, collect the like terms. Now your linear equation is in slope intercept form. From here you can read the location of the y-intercept from the equation, as well as solve for the location of the x-intercept by substituting 0 in for y in the equation. Let's look at an example of this process.
Here, we're asked to rewrite the equation y minus 0 equals 3 quarters times x minus negative 1 in slope intercept form, and to give the coordinates of its y-intercept. So let's look at this point-slope equation: equation y minus 0 equals 3 quarters times x minus negative 1. In this form, we can tell that graph of this equation passes through the point negative 1, 0, and that it has a slope of 3 over 4.
Let's write this equation in slope-intercept, and see what information is revealed in that form. On the left-hand side of the equation, y minus 0 is just y. And on the right-hand side, let's get rid of that double negative, and make x minus negative 1 just x plus 1. Next, we need to distribute that 3 fourths coefficient over both terms inside the parentheses. Doing that gives us the equation y equals 3 fourths x plus 3 fourths, which is in slope-intercept form, y equals mx plus b.
In this form, we can see that the y-intercept is located at the point 0, 3 fourths. Additionally, we can tell that the slope of this line is 3 fourths. We can also easily find the x-intercept by substituting 0 in for y in our equation, like this. If we solve for x here, we find that x equals negative 1, so the x-intercept is located at the point negative 1, 0.
Let's head over to the whiteboard to look at a couple more examples.
This first question reads, "Rewrite the equation y minus 6 equals negative 3 times x minus 4 in slope intercept form. Give the coordinates of its y-intercept." Alright, well let's write that equation down: y minus 6 equals negative 3 times x minus 4. There are no double negatives that we need to deal with, so the first thing we're going to do is distribute this negative 3 across both of these terms inside the parentheses. Doing that gives us y minus 6 equals negative 3x plus 12. And now the only like terms to combine are the constant terms, and we'll do that by adding 6 to both sides of the equation. And that gives us y equals negative 3x plus 18. And this is in slope intercept form, y equals mx plus b. So there's our slope intercept equation. Next, we're asked to give the coordinates the y-intercept. Well, the y-intercept is going to be at 0, because y-intercepts always have an x-coordinate of 0, and then this value for b, which is 18. So our y-intercept is at 0, 18. Alright, let's look at one more example.
This one reads, "Rewrite the equation y minus negative 2 equals one half times x minus 6 in slope intercept form. Give the coordinates of its y-intercept." Again, let's write that equation down. That's y minus negative 2 equals one half times x minus 6. We do have a double negative here, so let's deal with that first. That becomes y plus 2 equals one half times x minus 6. Now we want to distribute this one half coefficient across both of these terms. That gives us y plus 2 equals one half x minus 3. Now all that's left to do is combine like terms, and the only like terms are the constant terms, so we're going to do that by subtracting 2 from both sides of the equation, and that gives us y equals one half x minus 5. And this is in slope-intercept form, y equals mx plus b. Next, we want to find the coordinates of the y-intercept. Again, the x-coordinate of a y-intercept will always be 0, and the y-coordinate is going to be this b value, which for us is negative 5. So the y-intercept of this equation is located at 0, negative 5.
Question
When you rewrite an equation from point-slope into slope-intercept form, which intercept is revealed, the \( x \)-intercept or the \( y \)-intercept?
The slope-intercept form of a line's equation reveals the coordinate of the \( y \)-intercept.
