Hey, everyone. So we know how to solve linear equations. But once we solve a linear equation, we can actually then place it in a category. There are 3 possible categories that linear equations can be put into based on how many solutions they have. Now these solutions may be written as a solution set, which thinking back to set notation, this would just be our solution written between curly brackets, and that would be our solution set. Let's go ahead and jump into an example. So, we want to solve and then categorize these linear equations.

Looking at my first example, I have 2x+4=10. So my first step here is going to be to get all of my constants on one side and all of my x terms on the other. To move that 4 over, I need to go ahead and subtract it from both sides. So it'll cancel over here and I'm going to be left with 2x=10-4, which gives me 6. My next step here is to isolate that x variable. So, to do that here, I'm going to divide by 2 on both sides. Remember, whatever I do to one side of the equation, I have to do to the other. So, my 2 is going to be canceled, and I'm left with x=62, which is 3. This is my solution. For this particular linear equation, I only have one solution. The only value that would solve this that I could plug back in to get a true statement is 3. So my solution set is simply that number 3 in curly brackets.

Now all of the linear equations we've seen up to this point have had one solution, and their solution set would just be written as whatever number I get for x inside of those curly brackets. When I only have one solution, this is called a conditional linear equation. The reason it's called conditional is that it is only true on the condition that x equals some number. So for this particular linear equation, it is only true on the condition that x is equal to 3.

Looking at my next linear equation, I have x+5=x+2+3. Let's simplify that. On this side, I just have x+5 and then equals x. The 2 and the 3 are both constants, so I can combine them. 2 and 3 together are going to give me 5. So you might see where this is going but let's take this a step further. To move all of my constants to one side and all of my x terms to the other, I need to go ahead and subtract my 5 over here to cancel it out. Whatever I do to one side, I have to do to the other. And then to move my x over, I would need to subtract x from both sides. What happens here is I then have x minus x, which is 0, equals 5 minus 5, which is also 0. I just get 0 equals 0, which is an undeniably true statement. 0 is always equal to 0. This tells us that I could plug in any value for x at all and end up with a true statement. So I actually have an infinite number of solutions here because no matter what value I plug in for x, it will always be true. So that means that my solution set is all real numbers. All real numbers, I could plug back into my equation and get a true statement. And we denote real numbers or all real numbers with this fancy R just with an extra line in it. When I have an infinite number of solutions, this is called an identity equation.

Let's look at our final possibility. I have x=x+4. To get all of my x terms on one side with all of my constants on the other, I need to subtract x from this side. It will cancel here. Whatever I do to one side, I have to do to the other. So I'm left with x minus x. This gives me 0. Then on the right side of my equation, I'm left with 4 so I end up with 0 equals 4. Now this is definitely not a true statement. What this tells me is that no matter what value I plug in for x, it's just going to end up giving me something a little outlandish, like 0 equals 4, which we know is not true. So I actually have no solution here. There is no value that I could plug in for x to make this statement true. So my solution set is actually going to be written still in curly brackets with a 0 with a slash through it, and this is called the empty set because there is nothing inside that set. When this happens, when I have no solutions, this is going to be called an inconsistent equation.

So we have three types of linear equations. Our first one was conditional, when we have one solution. When we have infinite solutions, we have our second category, which is an identity equation. And then my very last one, if I have no solutions, that's my third type. It is an inconsistent equation. That's all for this one. Let me know if you have questions.