This blog will show you how to find the equation of a line in point-slope form given various different pieces of information about the line. To complete this problem, we’ll identify the components we’ve been given, plug them into the point-slope formula, and then simplify the equation. For expert training on calculus, enroll in the Become a Calculus 1 Master course from Udemy.
Let’s take a look.
Essentially, we have classified the information that you could be given into three different groups. We’ll look at each group to figure out how you would find the equation of the line and express that equation in point-slope form.
Before we look at the three groups, let’s examine that the equation of the line in point-slope form. This is the equation:
The equation can be expressed as y minus y sub 1 is equal to m times x minus x sub 1. Basically what you have to think about with this formula is that m is the slope of the line. That x, y will remain in your function. And that x sub 1 and y sub 1 represent a point on the line that we are given. So when you are given a point, you need to plug that point in for x sub 1 and y sub 1. That’s why it’s called the point-slope form, because you’re plugging in for the point – x sub 1 and y sub 1 – and the slope, m.
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Using the Point Slope Form Equation
Let us look at instances where the information you are given about the line includes one the following groups of information:
if you’re given the slope and the y-intercept
if you’re given the slope and a point
if you’re given a horizontal line
if you’re told that the line is the horizontal line through a point
if you’re given the horizontal line through a y-intercept
The above examples are all similar because you’re being given a slope and point, no matter which of these you’re given. So the approach to the point slope form equation is really similar for the above examples.
That’ll be slightly different than if you’re given two points on the line, which is its own group and which we will explain later. And that’ll be slightly different than being given a point on the line and being told that the line is parallel to another line. We’ll talk about each of those.
The Slope and the y Intercept or the Slope and a Point
If you are given the slope and y-intercept for example then being given the y-intercept is the same as being given a point. Say, for example, we’ve been told that the y-intercept is three. The x is 0 because it’s on the intercept so the point is (0,3), because x is equal to zero when the graph intersects y axis. If you’re given a slope and a y-intercept, and the y-intercept is three, then the point really is just (0,3). Obviously, that’s the same as a slope and a point. If you’re given a slope and a point, it’s really the same thing as being told the y-intercept. In either case, you’re still given a slope and a point.
A horizontal line or a horizontal line through a point
The same thing goes for the next two sets of information – horizontal line through a point or the y-intercept. The slope of any horizontal line is always zero. Essentially, you’re being told that the slope is zero and given a point, or that the slope is zero and given the y-intercept, which is also just a point.
So regardless of which of these pieces of information you’re given, you’re going to go about finding the equation of the line in the same way.
Let’s go ahead and look at how we could be given these different pieces of information and still get to the equation of the line in the same way.
Example 1: The Slope and the y Intercept.
If we’re given the slope as -2 and the y-intercept as 3, then you would use the following steps to calculate the point slope form equation.
Remember we need to find the elements of the following equation:
So we need to identify:
m (the slope)
First we need to realize that the y-intercept being 3 means that we have the point (0,3) so (x1,y1) is (0,3). What we’ll do is we’ll say, y minus – and then y coordinate of the point (0,3) – which of course is 3, is equal to m – the slope, which we know to be -2 – times x minus the x coordinate of the point that we’re given.
Remember that we’re given the point (0,3) because this is the y-intercept, so 0 is our x coordinate there. Notice how we plugged in the slope and the point (0,3). This has been plugged into the point-slope form, and we could leave it like this except we have this 0 here.
So we could simplify it to say, y minus 3 is equal to -2x. And we could even change this into y equals -2x plus 3, which is now slope-intercept form. Either way.
Obviously we can do the same thing if we’re told we have a slope and a point , -2 and (0,3), or even if we were given the point here, (1,1), which is also on the line.
Let’s assume we’re given the point (1,1) instead of (0,3). We could say y minus 1 is equal to -2 times x minus 1:
We plugged in our x coordinate, 1, here, our y coordinate, also 1, here, and our slope, -2.
When we do that, we can leave it in this form, or we can choose to simplify and get -2x plus 2. If we add 1 to both sides, we’ll get y equals -2x plus 3, and notice we’re back at the same place that we ended up with when we got the other set of information:
That’s how you deal with it if you have a slope and a y-intercept or a slope and a point.
Are you a visual learner? Check this out for point-by-point video explanations of point slope form equation.
Example 2: If you’re told that the line is the horizontal line through a point
If you were told that it was a horizontal line, you would know that the slope is 0. You’d plug in 0 for m, and then you’d plug in whatever point you have for x sub 1, y sub 1. If you’re given two points, as we are here, the points (0,3) and (1,1), you’ll need to use them first to find m.
The way that we’re going to do that, the equation for m is actually y sub 2 minus y sub 1, divided by, x sub 2 minus x sub 1.
What you want to do is you want to take one point – it doesn’t matter which point you call x sub 1, y sub 1 and which one you call x sub 2, y sub 2 – but let’s say that (1,1) here is x sub 1, y sub 1.
And that (0,3) here is x sub 2, y sub 2.
If we just plug those values in, y sub 2 is 3. Y sub 1 here is 1. X sub 2 we know to be 0, and x sub 1 we know to be 1. So we’ll get 2 divided by -1, which is a -2. That’s how we find the slope, m.
Now we’ll take the slope, m, and either one of these points – it doesn’t matter which one – and we’ll plug them into our point-slope formula. We’ll get y minus – let’s go ahead and plug in 1 and 1 again, (1,1), doesn’t matter again – but y minus 1 equals, our slope that we found, -2, times x minus 1.
And again, it’s the same thing we ended up with here. We can just simplify to -2x plus 2, if we want to. And if we want to, we can add 1 to both sides and transform it immediately into slope-intercept form, if we want to. This top line here represents point-slope form, the bottom line represents slope-intercept form.
Example 3: You are given a point and a parallel line
Lastly, if you’re given a point and the fact that the line is parallel to another line, essentially you’re being given the point here, and again, a slope. So this equation, y equals -2x minus 6 is in the slope-intercept form, which is y equals mx plus b – m being the slope. So -2 is the slope of this other line. Remember that lines that are parallel have the same slope. Essentially we’re being given a point and a slope, and we’re right back here into group one. So you just have to pick the slope out of this equation and then say, y minus our y coordinate here, 3, is equal to the slope here, -2, times x minus our x coordinate here, 0. And then again, you can leave it like this or you can simplify, -2x, and then if you want to, y equals -2x plus 3.
Calculus can be simple!
Notice how no matter what piece of information we’re given, we can quite easily get the point-slope form of the line, and we can quite easily transform any of those into the slope-intercept form of the line. Regardless of which method we use or what information we’re given, we always end up with the same result. For further discovery or review, Become a Calculus 1 Master by signing up for the course from Udemy today!