0. Review of College Algebra
Solving Linear Equations
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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. So 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 in 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. So looking at my first example, I have 2x+4 equals 10. So my first step here is gonna be to get all of my constants on one side, all of my x terms on the other. So 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 equals 10 minuteus 4, which gives me 6. My next step here is to isolate that x variable. So 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 equals 6 divided by 2, which is 3. So this is my solution. And 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 so their solution set would just be written as whatever number I get for x inside of those curly brackets. Now when I only have one solution, this is called a conditional linear equation. And the reason it's called conditional is because 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. So it's a conditional equation. Now all of our linear equations up until this point, like I said, have only had one solution. So let's take a look at some other possibilities. So looking at my next linear equation, I have x+5equalsx+2+3. So let's go ahead and simplify that. So on this side, I just have x+5 and then equals x. And then this 2 and this 3 are both constants, so I can go ahead and combine them. 23 together is gonna give me 5. So you might see where this is going, but let's take this a step further. So if I wanna 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. So what happens here is I then have x minus x, which is 0, equals 5 minus 5, which is also 0. So I just get 0 equals 0, which is an undeniably true statement. 0 is always equal to 0. So 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 actually all the real numbers. So 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. So when my when I have an infinite number of solutions, this is actually called an identity equation. So it's an identity equation because it's always going to be true no matter what. Infinite solutions, identity equation. Let's look at our final possibility. So over here, I have x equals x+4. So if I wanna go ahead and 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. So what I'm left with here is just a completely false statement. So what this tells me is that no matter what value I plug in for x, it's just gonna end up giving me something a little outlandish, a little crazy, 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 gonna 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. There is no number that I could plug into that equation to make it true. Now when this happens, when I have no solutions, this is going to be called an inconsistent equation. And you might also also hear it called as a contradiction because it gives you something crazy like 0 equals 4 or 7 equals 10 or something that you know is not true. So this is the last type or last category of our linear equations. Our first one was conditional. And when we have one solution, that is a conditional linear equation. Now when we have infinite solutions, we have our second category, which is an identity equation. And then my very last one, if I have have no solutions, that's my 3rd type. It is an inconsistent equation. That's all for this one. Let me know if you have questions.
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