Hey, everyone. In the previous chapter, we worked with linear expressions, some combination of numbers, variables, and operations like 2x+3. Now if I take that linear expression and I say that it needs to be equal to something, let's say this 2x+3 needed to equal 5, I now have a linear equation. Now equations are going to be a really important part of this course because we're going to have to solve a bunch of different types of equations. But don't worry, we're going to use a lot of the skills that we learned from linear expressions, and I'm going to walk you through solving these step by step. So like I said, a linear expression, if I just take it and add an equal sign, it is now a linear equation. Now it's called a linear equation because it is equal to something. If it's equal to something, it's an equation. When we worked with linear expressions, we would always simplify or evaluate it for some known value of x. We would be given that x is equal to a particular number like 4, and we would simply replace x in our expression with that value and get an answer. Now with linear expressions, we don't know what x needs to be, and we're tasked with solving for that unknown value of x. We have no idea what it has to be, but we want to find our value of x that makes our statement true. So for this particular equation, I would want to find a value of x that makes 2x+3 equal 5. Now, if I were to just guess what this x needed to be, let's say that that's how I wanted to solve it, I might guess that x is equal to 0. I could check that by saying 2 times 0 plus 3. But 2 times 0 is 0. I would just get 3. That's not equal to 5. So x equals 0 wouldn't be my answer here. Then I could go and say x equals 1, maybe. So if I plug that in, 2 times 1 plus 3, 2 times 1 is 2, plus that 3, this actually does give me 5, which is great. But we probably don't want to do that for every linear equation that we're getting because that would honestly just become really annoying. So, we're going to need to use some other skills here in order to solve our linear equations. So, you're actually going to need to use all of the different operations in your toolbox, addition, subtraction, multiplication, and division, in order to isolate x to get it by itself. Now these operations should always be done to both sides of the equation. This is super important, and it's going to be important throughout this course. Whatever you do to one side of the equation, you have to do to the other. So let's get some practice with isolating x. In this example, we want to identify and perform the operation needed to isolate x by applying it again to both sides. So looking at my first example, I have x plus 2 equals 0. Now this 2 is being added to the x. So how do I get rid of it in order to get x by itself? Well, the opposite of adding 2 would be subtracting 2. So to get rid of it, I could subtract 2, and I need to do that to both sides. So my left side here would cancel, and I'm just left with x equals 0 minus 2 gives me a negative 2, and I have isolated x. Looking at our second example, I have 3x equals 12. Now here, my 3 isn't being added to the x. It's actually multiplying it. So what operation could I do in order to get rid of that 3 that's multiplying my x? Well, the opposite of multiplication is division, so I could go ahead and divide by 3 on both sides to cancel it out. And, again, I'm just left with x equals 12 divided by 3 gives me 4. So I've isolated x here. Now you might have noticed that for these, we were doing opposite operations. If ever I want to get rid of something in order to isolate x, I'm always going to do the opposite operation of whatever is happening in the equation. So when we saw something being added, the opposite operation to get rid of that was subtraction. And when we saw something being multiplied, the opposite operation was division. Now this, of course, also works the other way. So if I see something being subtracted, I could add it in order to get rid of it. And if I see something being divided, I could always multiply to get rid of it. But for these, we only saw one operation needed in order to isolate x, and you're often going to actually have to do multiple operations in order to solve a linear equation. So let's take a look at that. In this example, I want to solve the equation 2 times x minus 3 equals 0. So let's go ahead and take a look at our steps. Our very first step is to distribute our constants. And looking at my equation here, I have this 2 that needs to get distributed to both the x and the negative 3. So if I do that, I get 2x, and then 2∗−3 gives me negative 6 equals 0. So step 1 is done. Now step 2 asked me to combine like terms. So taking a look at my equation, I have 2x−6 equals 0. I don't have any like terms here because I just have an x term and I have a constant. I can't combine that any further. So step 2 is also done. Now in step 3, we want to group terms with x and our constants on opposite sides. Now it doesn't matter which side I put my x terms and which side I put my constants on as long as they're on opposite sides. So let's go ahead and do that. I have 2x−6 equals 0. So I want to pull this 6 over to get it on the opposite side. So in order to do that, I need to do my opposite operation here, which is going to be adding 6 to both sides. It will then cancel there, and I'm left with 2x equals 0 +6 gives me 6. So step 3 is done. Now step 4 is to isolate x. You might also hear this called solve for x. So let's go ahead and isolate x. I have my 2 multiplying my x here. So that means to get rid of that 2, I need to divide. So if I divide both sides by 2, my 2 will cancel here. I'm left with x equals 6 divided by 2 gives me 3. And this is actually the answer. So I'm done with step 4. And 3, my solution here or my answer here is called the solution or the root of the equation. You might hear it called either an answer, a solution, or a root. Any of these are referring to the same thing. So we actually do have one more step here, step number 5, and that is to check our solution by replacing x in our original equation. So I'm going to take my original equation, 2x−3 equals 0, and make sure that I found the value of x that makes that true. So I take 2, the value I got for x was 3 minus 3 equals 0. Now this then gives me 2, 3 minutes 3 is 0, equals 0. And we know that anything times 0 is 0, so I am left with 0 equals 0, which is definitely a true statement. So I've completed solving this linear equation. That's all for this one, guys. Thanks for watching.