Okay, in this brief segment, I have three examples, and I've titled it More Linear Equations. A lot of times when I'm working with students, I tell them, this is some of the kind of weird cases. You just got to be on your game here and make sure you recognize them.
They're not extra difficult. In fact, these probably are a little bit easier. But I've got three examples. If you want to pause the video and write them down and follow along with me, that'd be great. So let's begin.
This first example, what I notice is that I could combine these terms on the left and just get 5x. Now, typically what I do is I bring all the x's to one side. So I usually bring them to the left. Again, it doesn't matter which side you bring them to. But you notice on the right-hand side, I just end up with 0. 0 is a legitimate number.
On the left, 5 minus 8 is negative 3. So I have negative 3x. If I follow my pattern here, I just divide off the negative 3. And so that scratches out those and just becomes 1x. And 0 thirds or negative 0 thirds, it really doesn't matter here. 0 divided by 3, 0 thirds.
If somebody says, how many thirds do you have? I have 0 thirds. That means I don't have any thirds.
So the answer is just 0. So pause a second. Let your common sense help you with this. If you let x be 0, 4 times 0, you know that's 0, plus 0 equals 0. So it's a legit answer.
It's fine. The answer is 0. Equation number two. Looking at it, you say, well, it looks like I can practice some distributing.
Again, distributing just means multiplying. So I end up with 2x plus 4 plus a 2x. Over here, I've got 4x plus 4. Again, no worries, just taking my time through this thing.
I can combine on this side over here to make it a little simpler. The two x's make four x. Don't look too tightly at this.
I'm going to try to get the x's on one side. I'm going to subtract, just like I've been doing in all these other examples. And then notice over there, I don't have any X's. So that's okay.
You get zero right there. So you've got a four on the left side. You've got a four on the right side.
You say, where's my X? I say, well, it's gone. No big deal. The question is, does four equal four?
And of course, you know, the answer to that is true. But a lot of times what I'll do in my notes is I'll just say, look, this is a true statement. And so I say, if it's true, then that means, let me just write this out for you, since that is true, that means that x can be any value, any number. You get real fancy.
We can, in math class, we can say it can be any value, or we can say it can be any real number. Hey, but if that's too much, Too much math for you right there. Just say x can be anything because I got a true statement.
So you say, can x be 1? Yeah. Put your 1 in there and go check it. It's true. Can x be a negative 50?
Yeah. Can x be a negative 3 7ths? Yes.
And you can just check them one after the other. I won't spend video time on that. So you say, what's my answer? Any real number.
or x can be anything. There's my answer. Now let's contrast that with this last example here.
Say I'm going to distribute just like, you know, and I don't see anything over there to do, so let me pause and distribute 4x plus 20. And over here I'm just going to write it down. Put my X's together. I know you can see something weird's about to happen.
That's okay. My goal is I'm just trying to get the x's on one side. And you say, well, you just got rid of them all.
Well, it couldn't help that. So I end up with a 20 on the left side, a 20 on the right side. 20 doesn't equal 21. That's a false statement.
Since it's false, then There is no solution. Let me write it this way. There is no value for x that will work.
I'm just trying to put it in real plain, simple words. There's nothing you can put in there for x. A lot of times we'll just say no solution. Fancy mathematical terms is you can draw this.
This means it's an empty set. A set is just a bag of numbers that can work in that thing and there's nothing in that bag that'll work. So no solution, no solution, another way of writing that, just simply saying, you know, there's no value of x that's going to make this thing true because my x is left. So I paused there to give you this short video on just here are some cases that I see. I see students sometimes get mixed up here, so just make sure you follow through the steps.
What happens if it's true? My variables are gone and it's true. What happens if my variables are gone and it's false? So hang on, we're going to do one more review of these equations.