Writing a system of linear equations from tables
About this video
The video writes a system of equations from two tables of points. For each line it finds the slope from two points, uses point-slope form, and rearranges into standard form, giving line n: x + 2y = 5 and line p: 3x + y = 2.
You'll be able to
- find the slope of a line from two points in a table.
- use point-slope form, y - y1 = m(x - x1), to write the equation of a line.
- rewrite a linear equation in standard form and combine two equations as a system.
Key words
- system of equations
- Two or more equations considered together, here one for each line.
- slope
- Change in y divided by change in x between two points on a line.
- point-slope form
- y - y1 = m(x - x1), using a slope m and a point (x1, y1).
- standard form
- A linear equation written as Ax + By = C, like x + 2y = 5.
Transcript
- [Instructor] We're told, "The tables below contain points on the graphs of line n and p." All right, "Write a system of equations in standard form that represents lines n and p." So we just have to write a system of equations, we don't have to solve it. So pause this video and have a go at that before we do this together.
Okay, well, in order to write the systems of equations, we just really have to write the equation of each of these lines. And if we wanna write it the way a system is typically written, we can write them in standard form. So it really boils down to, can we figure out the equation of line n and can we figure out the equation of line p, and then can we put them in standard form?
So to start off, let's just figure out the slope of line n. So the slope here, remember, slope is just going to be change in y over change in x. And if we just take these first two points right over here, our change in y is equal to, if I go from three to one, my change in y is negative two.
And if I go from negative one to three, what's my change in x there? My change in x is going to be, I just increased by four, so this would be equal to negative two, my change in y if I go from three to one is negative two, over my change in x, if I go from negative one to three, that's four.
So it's equal to negative 1/2. And if this is indeed a line, I should get the same thing no matter which two points I pick. For example, if I picked these second two points, my change in y, change in y, is negative two minus zero, if I do the endpoint minus the starting point, over change in x, my change in x is going to be nine minus five.
Once again, negative two over four. I didn't have to do that. I'm just showing you that for a line, I can pick any two points and the slope between them should be the same if they all sit on a line. So now we're close, we have the slope, but we haven't leveraged the points that we are given here.
And so what we can do is leverage point slope form of a line to write that down. So point slope, what we wanna do is write y, maybe I'll do that original blue color, y minus one of, we pick one of the points. And I mean, let's pick, actually, well, we could pick anything.
Let's actually pick this one. This one could be interesting. So if we pick y minus this actual point, this actual y-coordinate, zero, is going to be equal to our slope, negative 1/2, times x minus the corresponding x-coordinate of that point, x minus five. And now we can just algebraically manipulate this into any form we want.
But I wanna put it into standard form. So this is just gonna be the same thing as y is equal to, if I distribute this negative 1/2, I get negative 1/2x, and then I have plus 5/2. now there's a bunch of things I could do here. Maybe a simple thing I could do is, if I just don't wanna have all these fractions laying around, I could multiply everything by two, I could get 2y is equal to negative 1x, or I could just say negative x plus five.
And then if I want, I could add x to both sides. And so I would get x plus 2y, x plus 2y is equal to five. And that's this first line, line n, in standard form. Now let's move on to line p, which I will do in orange. Same idea, what is our slope? Our slope, and I'll just write it out here, slope is going to be equal to our change in y.
So let's just look at these two points over here. They're the first two. Our change in y going from five to negative one is negative six. You could also do that as just negative one minus five is negative six. And then one minus negative one is two, or another way to think about, if you're going from x equals negative one to x equals one, you're increasing by two.
So our slope here is equal to negative three. And so let's just use this first point over here. I could use any of these four points, put it in point slope form. I could say y minus five is equal to our slope. negative three times x minus negative one, minus negative one is the same thing as plus one.
And so I get y minus five is equal to negative 3x minus three. And then I could add five to both sides of this equation, and I would get y is equal to negative 3x. If I added five to both sides, this five goes away. And then this is going to be plus two. And then I could add 3x to both sides, and I would get 3x plus y is equal to two.
So that's our second equation. And if I just wanted to write it as a system, just to make it clear, I'll do that over here. The system would be line n is x plus 2y is equal to five, and then line p is 3x plus y is equal to two. So that right there is my system of equations.
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