Writing the Equation of a Line Given Two Points | Slope-Intercept Form (y=mx+b)

[Music] Welcome to Math with Mr. J. In this video, I'm going to cover how to write an equation in slope intercept form from two points. We will go through some examples together here and then I'll give you some practice problems that you can try on your own. Let's jump into number one where we are given two points on a line 54 and 106. So our line goes through those two points. We need to use those two points to write the equation of the line in slope intercept form. Now this may seem confusing at first, but after our examples, this is going to make more sense. We're going to use the three steps off to the side. Our first step, we need to find the slope of our line. Remember, slope equals rise over run. the vertical change over the horizontal change. When given two points, we can use this formula right here to find the slope of the line. So slope equals y2 - y1 over x2 - x1. So we are using the x and y coordinates of our points. Y2 - Y1 gives us the rise and X2 - X1 gives us the run. 54 is going to be our first point and 106 is going to be our second point for our first point. 54 5 is our x coordinate. So this is going to be x1. Four is our ycoordinate. So 4 is going to be y1. And it is helpful to label these like we are right now. Our second point 106 10 is our x coordinate. So 10 is x2. 6 is our ycoordinate. So 6 is y2. Now all we need to do is plug those in to find our slope. So slope equals, and I'll use m for slope here, y 2 - y1. So 6 - 4 over x2 - x1. So 10 - 5. 6 - 4 gives us 2. And 10 - 5 gives us 5. So the slope of our line here is 2 fths. Now we can move on to step two. We need to find the y intercept. We use that slope and then one of our points to figure out the y intercept. Remember for slope intercept form, y = mx + b. So I'm actually going to write that out off to the side here. So y = mx + b. Remember m is the slope, b is the y intercept, and then x and y are the coordinates of any point on the line. So we need to plug in two fths for m. And then we need to use a point on the line and use those coordinates for x and y. We're going to use 54. Now it does not matter which point we use. We can use 10 six. We just need to pick one. So again we're going to use 54. So what we do? We plug in four for y our ycoordinate. So four equals then we plug in two fifths our slope for m. So 2 fths * x which is 5 our x coordinate from our point. So plug in five plus b. b is what we're solving for. We're looking to solve for that y intercept. The first thing that we need to do here is multiply. We need to do two fths * 5. Now let's come to the top here and multiply. So we have 2 fths time 5. And I'm writing five in fractional form since when we multiply fractions we multiply straight across. So we need to do 2 * 5 which is 10 over 5 * 1 which is 5. So we end up with 10 fths which is an improper fraction that we can simplify here and do 10 / 5 that gives us 2. So 2 fths * 5 is 2. So we have 2 + b equals and then on the left side we have four. So now we need to isolate b. Here we are adding two to b. So in order to undo that, we need to do the opposite, which is subtraction. So let's subtract two from the right side of the equation. Whatever we do to one side of the equation, we must do to the other. So subtract two from the left side as well. The twos on the right cancel each other out. So b is now isolated. b equals and then on the left we have 4 - 2 which gives us 2. So b = 2. That's our y intercept. This line crosses the y ais at two at 02. So now we have our slope from step one and our y intercept from step two. That's everything we need for our equation. So now for step three, we write our equation. We plug in the slope and y intercept. So we have y equals and then we plug in two fths for m our slope x plus our y intercept of 2. And this is our equation. And that's it. We are done. All this basically boils down to and this is how I like to remember it is slope and solve. So we find the slope and then solve for the y intercept. Then we are good to go to write our equation. Now our three steps off to the side those are a little more detailed but thinking slope and solve can give us a good starting point as far as thinking through these problems. Let's move on to number two. Moving on to number two. Here our two points that our line goes through. First point -21. Second point 1 -8. Let's start by finding the slope. So our first point -21 -2 is x1 and then 1 is y1. Then our second point 18 1 is x2 and8 is y2. So m equals and then we have y 2 which is -8 - y1 which is 1 over x2 which is 1 - x1 which is -2. Now we subtract here. So we have -8 minus 1. And remember whenever we have subtraction problems involving negatives, it can help to add the opposite. That's going to give us the same thing as subtraction. So for -8 - 1, let's add the opposite here. So we end up with -8 + -1. That gives us -9 over. And now we have 1 - -2. Let's add the opposite here and we end up with 1 + 2. That gives us three. So we have a slope of -9/3 -9 over3 which we can simplify. And we need to do -9 / 3. That gives us -3. And that's our slope. Now we need to use that slope to find the y intercept. So moving on to step two. Let's use that slope and one of our points. Let's use -2 1. So we have y equ= mx + b. So let's plug in now. And we'll start with our ycoordinate of 1. So plug in one for y equals mx. So our slope of -3 * our x coordinate of -2 + b. So we need to solve for b here. And the first thing that we need to do is -3 * -2. A * a negative gives us a positive. So this gives us a positive 6. So 6 + b equals and then we have 1 on the left side. Now we need to isolate b. 6 is being added to b. So let's subtract six. Subtract a positive six. Whatever we do to one side of an equation, we must do to the other. So subtract six from the left side as well. The sixes on the right cancel each other out. B is now isolated. equals and then we have 1 - 6 on the left side of the equation. 1 - 6 gives us -5. So our y intercept is -5. Our line crosses the y ais at -5. And now we have all the information we need to write our equation. So y equ= our slope of -3 x plus our y intercept of -5. Now this equation is correct but we're actually going to simplify it so we don't have that double sign the addition and then the negative. So we can rewrite this using subtraction. y = -3x - 5. Now, a lot of times we will see slope intercept form simplified to show a negative y intercept as subtraction. So, something to keep in mind. So, this equation will work right here where we add -5 or we can simplify that and rewrite it as subtraction. So, either of these equations will work. So, now that we've gone through some example problems together, it's time for you to try some practice problems on your own. So, have paper and a pencil ready. So, here are your two practice problems. Use the two points given for each problem here to write the equation of each line in slope intercept form. I'll give you five minutes and then we will go over the answers. Feel free to press pause if you need more time or jump ahead to the answers whenever you are ready. Go ahead and start. [Music] Heat up. [Music] [Music] Nat. Hey. Hey. [Music] Hey, hey, hey. [Music] [Music] [Music] [Music] Hallelujah. [Music] [Music] [Music] Okay. So that was 5 minutes. Let's go over the answers. So here are our answers. For number one, the slope is 34s and the y intercept is -2. So our equation is y = 34 x + -2 or y = 34 x - 2. For number two, the slope is -4 and the y intercept is 7. So our equation is y = -4x + 7. So there you have it. There's how to write the equation of a line from two points on the line. I hope that helped. Thanks so much for watching. Until next time, peace.