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 the consistent system of equations has at least at least one solution at least one solution and an inconsistent system of equations as you can imagine has no solutions has no solutions so if we think about it graphically what would a what would the graph of a consistent system look like well either 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 they would intersect at a ton of points actually 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 it 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 would have a different y-intercept so it would look like this so that would 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 let me draw my x-axis let me draw my y-axis 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 on each of 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 zero when X is zero you have two y is equal to 13 so when X is zero you have two y is equal to 13 or Y is equal to 13 over to 13 over 2 which is the same thing as 6 and 1/2 6 and a half 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 over 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 I'm 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 that 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 you get knit this 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 when y is 0 you have you'd 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 over 3 negative 11 over three and this is the exact same thing as three negative three and two thirds so when y is zero you have you have X being negative three and two-thirds so it would be so maybe this is about six so negative three and two-thirds would be right about here so this is the point negative eleven thirds comma zero and so the second equation will look like something like this will look something like that now clearly and you know that 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 equation it has one solution you just have to have at least one in order to be consistent so once again consistent system of equations