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System - Number of Solutions
In the last lesson, in fact, the last couple lessons, I said that a system of two equations with two unknowns usually has a single unique solution. Altogether, there are three cases. There’s the case of one unique solution for x and y. There’s the case of an infinite number of solutions, then there’s the case of no solution possible, no solution at all.
Now, in the previous lessons, we discussed case one already, we discussed that in great detail. How to find the solution, the single solution either by substitution or elimination. I will remind you, that case corresponds to a case in which two lines on the x-y plane intersect at a single point.
So usually, you pick one random line, and another random line, and what happens is, you’re just gonna get those lines intersecting at a single point. So that is by far the most common case, but it’s not the only case, and there are two other cases possible. Case two is the infinitely many solution. So, I’ll say pause the video and try to solve this system on your own.
Okay, let’s talk about this. The first equation has a negative y, so this is an easy candidate for substitution. We get y equals 2x minus 5. Very good. Plug this right into the second equation, very good, then distribute, then cancel, and wait a second.
All the x’s went away. What exactly is going on here? So, if you get to a point where the x’s suddenly disappear, suddenly there are no variables in there, just numbers. Don’t automatically assume that you’ve done something wrong. On this page, every single step I followed was mathematically correct, and it wound me up here at negative 10 equals negative 10.
So what exactly is going on here? We started with the system, and ended up with an equation, negative 10 equals negative 10. In other words, an equation that is always true. It doesn’t matter what x or y equal, it’s always gonna be true, that x equals, that negative 10 equals negative 10.
That is true 100% of the time. So it is a truth not dependent on the values of x and y. Well, that means if we wind up with something that’s always true, we must have started with something that was always true. Means that the starting point, the system must be always true. That is true for all values of x and y.
No matter what we make the values of x and y. We will have a system that works. We’ll have a true system. How did this happen? Let’s go back and look at the original equations. Rewrite the second equation, we’ll just put the x first.
Now multiply the first equation by negative two, and lo and behold. It’s the same equation. One equation is just a multiple of the other so really just two copies of the same equation. At the beginning of the video, on two equation and two unknowns, we discussed how a single equation with two variables would have an infinite number of solutions.
That’s exactly what we have in this system. Cuz, really, we have, it looks like two equations are written on the page, but really it is just two copies of the exact same equation. We really have only one equation. And one equation, as we know, two variables, one equation, we have an infinite number of solutions.
In a way, we are asking, where does a line intersect itself? It’s as if we picked two random lines and by the most unlucky chance imaginable, we just happened to pick the same line twice. And so it lands on itself and intersects itself in an infinite number of places. If you notice that one equation is a multiple of the other, that means you are in the infinitely many case.
Sometimes you can just spot that. Even if you don’t notice that and solve, you’ll wind up with an always true equation. You’ll wind up something with all the variables drop out, and you just get something like 7 equals 7, something that’s always true. When you get something that’s always true, that’s an indication that the system of equations is also always true for all values of x and y.
So this is the infinitely many solutions case. Now let’s talk about the no solutions case. Pause the video and try to solve this system. Okay, much in the same as the last time, we have an x by itself, so that’s a very obvious candidate for substitution.
x equals 2y plus 5. Very good. Now plug that into the second equation, distribute, cancel. And what’s going on here? Again, all the variables have dropped out. But now we have something that is patently false.
It is never true that 15 equals 8. We’ve, we’ve run into a contradiction. Here we started with the system of equations and it led us to a never true equation. This means that the system of equations must be never true. In other words, there’s no possible values of x and y that we could pick that would allow us to say that 15 equals 8 is a true statement.
So the system does not work for any values of x, and there is no solution. How can this happen? Think of the original system again. Multiply the first equation by 3, and what we get is 3x minus 6y equals 15. And notice we have the same expression on the left in both equations equal to different things.
We have the same expression equal to two different things. That’s impossible. There’s no way that 3x minus 6y can equal both 8 and 15 at the same time. And that’s why we relate to the paradox 8 equals 15. In terms of the graphs, this would be the case of parallel lines. We pick two lines at random and just by pure chance, we happened to pick two parallel lines.
In a way, we are asking, where do the parallel lines intersect? The answer, of course, is that they don’t. Parallel lines never intersect. If you try to solve the equation and your work results in a never true equation, you wind up with something like 2 equal 7. Then it means that the system is never true.
It has no possible solutions. In summary, a system of two equations with two unknowns may have one solution, no solutions, or infinitely many solutions. The most common case, by far, is the one solution case. And that’s where you would use substitution or elimination and solve for individual values of x and y.
If you do that solving and you get an always true equation, something like 12 equals 12, then the system is always true and has infinitely many solutions. If you solve and all the variables drop out and you wind up with a never true equation, something like negative 2 equals positive 9, then the assist, then the system has no solutions.
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