Solutions to systems of equations: dependent vs. independent

Is the system of linear equations below dependent or independent? And they give us two equations right here. And before I tackle this specific problem, let's just do a little bit a review of what dependent or independent means. And actually, I'll compare that to consistent and inconsistent. So just to start off with, if we're dealing with systems of linear equations in two dimensions, there's only three possibilities that the lines or the equations can have relative to each other. So let me draw the three possibilities. So let me draw three coordinate axes. So that's my first x-axis and y-axis. Let me draw another one. That is x and that is y. Let me draw one more, because there's only three possibilities in two dimensions. x and y if we're dealing with linear equations. So you can have the situation where the lines just intersect in one point. Let me do this. So you could have one line like that and maybe the other line does something like that and they intersect at one point. You could have the situation where the two lines are parallel. So you could have a situation-- actually let me draw it over here-- where you have one line that goes like that and the other line has the same slope but it's shifted. It has a different y-intercept, so maybe it looks like this. And you have no points of intersection. And then you could have the situation where they're actually the same line, so that both lines have the same slope and the same y-intercept. So really they are the same line. They intersect on an infinite number of points. Every point on either of those lines is also a point on the other line. So just to give you a little bit of the terminology here, and we learned this in the last video, this type of system where they don't intersect, where you have no solutions, this is an inconsistent system. And by definition, or I guess just taking the opposite of inconsistent, both of these would be considered consistent. But then within consistent, there's obviously a difference. Here we only have one solution. These are two different lines that intersect in one place. And here they're essentially the same exact line. And so we differentiate between these two scenarios by calling this one over here independent and this one over here dependent. So independent-- both lines are doing their own thing. They're not dependent on each other. They're not the same line. They will intersect at one place. Dependent-- they're the exact same line. Any point that satisfies one line will satisfy the other. Any points that satisfies one equation will satisfy the other. So with that said, let's see if this system of linear equations right here is dependent or independent. So they're kind of having us assume that it's going to be consistent, that we're going to intersect in one place or going to intersect in an infinite number of places. And the easiest way to do this-- we already have this second equation here. It's already in slope-intercept form. We know the slope is negative 2, the y-intercept is 8. Let's put this first equation up here in slope-intercept form and see if it has a different slope or a different intercept. Or maybe it's the same line. So we have 4x plus 2y is equal to 16. We can subtract 4x from both sides. What we want to do is isolate the y on the left hand side. So let's subtract 4x from both sides. The left hand side-- we are just left with a 2y. And then the right hand side, we have a negative 4x plus 16. I just wrote the negative 4 in front of the 16, just so that we have it in the traditional slope-intercept form. And now we can divide both sides of this equation by 2, so that we can isolate the y on the left hand side. Divide both sides by 2. We are left with y is equal to negative 4 divided by 2 is negative 2x plus 16 over 2 plus 8. So all I did is algebraically manipulate this top equation up here. And when I did that, when I solved essentially for y, I got this right over here, which is the exact same thing as the second equation. We have the exact same slope, negative 2, negative 2, and we have the exact same y-intercept, 8 and 8. If I were to graph these equations-- that's my x-axis, and that is my y-axis-- both of them have a y-intercept at 8 and then have a slope of negative 2. So they look something-- I'm just drawing an approximation of it-- but they would look something like that. So maybe this is the graph of this equation right here, this first equation. And then the second equation will be the exact same graph. It has the exact same y-intercept and the exact same slope. So clearly these two lines are dependent. They have an infinite number of points that are common to both of them, because they're the same line.