Solutions to systems of equations: consistent vs. inconsistent

Is the system of linear equations below consistent or inconsistent? And they give us x plus 2y is equal to 13 and 3x minus y is equal to negative 11. So to answer this question, we need to know what it means to be consistent or inconsistent. So a consistent system of equations. has at least one solution. And an inconsistent system of equations, as you can imagine, has no solutions. So if we think about it graphically, what would the graph of a consistent system look like? Let me just draw a really rough graph. So that's my x-axis, and that is my y-axis. So if I have just two different lines that intersect, that would be consistent. So that's one line, and then that's another line. They clearly have that one solution where they both intersect, so that would be a consistent system. Another consistent system would be if they're the same line, because then they would intersect at a ton of points, actually at an infinite number of points. So let's say one of the lines looks like that. And then the other line is actually the exact same line. So it's exactly right on top of it. So those two intersect at every point along those lines, so that also would be consistent. An inconsistent system would have no solutions. So let me again draw my axes. Let me once again draw my axes. It will have no solutions. And so the only way that you're going to have two lines in two dimensions have no solutions is if they don't intersect, or if they are parallel. So one line could look like this. And then the other line would have the same slope, but it would be shifted over. It would have a different y-intercept, so it would look like this. So that's what an inconsistent system would look like. You have parallel lines. This right here is inconsistent. So what we could do is just do a rough graph of both of these lines and see if they intersect. Another way to do it is, you could look at the slope. And if they have the same slope and different y-intercepts, then you'd also have an inconsistent system. But let's just graph them. So let me draw my x-axis and let me draw my y-axis. So this is x and then this is y. And then there's a couple of ways we could do it. The easiest way is really just find two points on each of these that satisfy each of these equations, and that's enough to define a line. So for this first one, let's just make a little table of x's and y's. When x is 0, you have 2y is equal to 13, or y is equal to 13/2, which is the same thing as 6 and 1/2. So when x is 0, y is 6 and 1/2. I'll just put it right over here. So this is 0 comma 13/2. And then let's just see what happens when y is 0. When y is 0, then 2 times y is 0. You have x equaling 13. x equals 13. So we have the point 13 comma 0. So this is 0, 6 and 1/2, so 13 comma 0 would be right about there. We're just trying to approximate-- 13 comma 0. And so this line right up here, this equation can be represented by this line. Let me try my best to draw it. It would look something like that. Now let's worry about this one. Let's worry about that one. So once again, let's make a little table, x's and y's. I'm really just looking for two points on this graph. So when x is equal to 0, 3 times 0 is just 0. So you get negative y is equal to negative 11, or you get y is equal to 11. So you have the point 0, 11, so that's maybe right over there. 0 comma 11 is on that line. And then when y is 0, you have 3x minus 0 is equal to negative 11, or 3x is equal to negative 11. Or if you divide both sides by 3, you get x is equal to negative 11/3. And this is the exact same thing as negative 3 and 2/3. So when y is 0, you have x being negative 3 and 2/3. So maybe this is about 6, so negative 3 and 2/3 would be right about here. So this is the point negative 11/3 comma 0. And so the second equation will look like something like this. Will look something like that. Now clearly-- and I might have not been completely precise when I did this hand-drawn graph-- clearly these two guys intersect. They intersect right over here. And to answer their question, you don't even have to find the point that they intersect at. We just have to see, very clearly, that these two lines intersect. So this is a consistent system of equations. It has one solution. You just have to have at least one in order to be consistent. So once again, consistent system of equations.