Writing a system of linear equations from a graph

0:00- So over here it looks like we have two lines graphed on this coordinate plane. And we're told to write a system of linear equations in slope intercept form that represents this graph. So pause this video, have a go at that before we do it together. Okay. Now a system of linear equations, we're just going to essentially write the equations for each of these lines as a system.

0:21So let's just figure out the equation for each of these lines. So I'll start with this brown line here. And what we wanna do, we wanna write each of these in slope intercept form. And we could first just figure out the slope. So the slope, I could say that's M, that's going to be change in Y over change in X.

0:40And so, let's say we're going from this point on the left to the point on the right. So our change in X going from -4 to 4 is going to be equal to 8. That's how much we increased in X and our change in Y as we go from this left point to this right point, we were at Y equals 4 and we go to Y equals 2.

1:03So our change in Y is -2. And so our slope is -1/4. Now this line, it is tempting to just say, hey look, it looks like we intersect or our Y intercept is right over there at 0, 3. And if we assume that, we immediately could say, okay, this line is going to be Y is equal to our slope is -1/4.

1:25X plus our Y intercept plus 3. But we could validate that if we like, instead of just jumping to the conclusion and doing it visually, I could also say, alright, I know the slope is -1/4, so Y is equal to -1/4X, but I don't know the Y intercept yet. Let's say I wasn't able to see it so cleanly off of this graph and then I could take one of these other points and substitute in for Y and X to solve for B.

1:52So I could take this, say the second point, I could do it with either one, but let's say, let's see when X is 4, Y is 2. So 2 is equal to -1/4 times 4, X is 4, Y is 2 plus B. And so I get 2 is equal to

2:09- 1/4 times 4 is -1 plus B, add 1 to both sides and you get B is equal to 3, just like we had before. So that is the equation of the first line. Actually, let me not square that 'cause they're not just asking us for the equations. I'm gonna write it as a system in a bit.

2:25But now let's do that second one in that teal-ish color. So same idea. And this one we can't just eyeball the Y intercept, it doesn't even, we can't see it at least what we see graphed right now, it does at some point intersect to the Y axis, but we could see the slope.

2:41So for this one is, I'll just say slope is change in Y over change in X. And here our change, if I go from this left point to this right point, our change in Y, you could view it as 9 minus 1, or you could just count these, we go up by 8. So our change in Y is 8.

3:00When our change in X, it looks like we're going our change in X, we're increasing X by 4, we're going from X equals -7 to X equals -3, which would be 4. So 8/4, which is equal to 2. We have a slope of 2. And you can even see that here. If we go 1 to the right, we go up 2, we go 1 to the right, we go up 2.

3:24But now let's figure out it's Y intercept. So we know that this is going to be Y is equal to 2, our slope X plus B. And let's substitute one of these points in. I'll do this first one. So Y is equal to 1, when X is equal to -7,

3:38- 7 plus B. So 1 is equal to -14 plus B, add 14 to both sides and you get 15 is equal to B. And so our first... I'll just rewrite it, our first equation or this brown line here, that is Y is equal to -1/4X plus 3. And the second one in the teal color is Y is equal to 2X, 2X plus 15.

4:11And there you have it. I have written it as a system of equations where both of the equations are in slope intercept form.