Master Solving Linear Equations in Under 8min

[00:00:00]Solving linear equations, the fundamental skill all students need to know how to do. And even students that know exactly what to do, they still make the classic mistakes. So, in this video, we are going to master once and for all solving linear equations. We're going to go through one step, two step, as well as something you don't see in a lot textbooks, which is going to be the three-step process. So, let's go and start with the one step. And the one thing I want you to know about one-step equations is just identify what is being applied, what operation is being applied to your variable, and then you need to undo it. So, let's work through this. X plus four. You can just see as you read it, X is being or four is being added to X. So, we need to undo the addition. So, we're going to undo addition using the inverse operation, which is subtraction.

[00:00:44]And we need to use the property of equality, which is going to means apply that operation to both sides. So, we get an X equals -3.

[00:00:51]Over here, X minus two. Inverse operation is addition. Property of equality on both sides. X is going to equal a positive one. Over here, it's four times X, right? So, we need to undo multiplication, which means we're going to use the inverse operation, which is division. Divide a four on both sides, and we get X is equal to a positive three. Over here, you can see X is being divided by three, right? So, we need to undo division, which is means multiplication. So, you multiply three on both sides, and therefore we're going to get X equals Now, 27 times three, 27 times two is going to be 54. If you add one more 27 to it, that's going to equal a 81.

[00:01:28]Now, let's get into two-step equations.

[00:01:30]Now, the one thing I want you to understand about two-step equations is we always have to undo addition and subtraction before we undo multiplication and division. And we call them two steps is because there's usually going to be two operations that are applied, addition and subtraction as well as a multiplication and division.

[00:01:46]So, following the reverse order of operations, always undo addition and subtraction before multiplication and division. So, you can see here, this 2X or two times X is being added by one.

[00:01:56]So, I'm going to undo addition and subtraction first. Now, I get a 2x is equal to a positive 6. Undo multiplication, divide by 2, x is going to equal a 3. Over here, I have a -3x - 4. So, I'm going to undo my subtraction.

[00:02:08]So, I'm going to add a 4 on both sides.

[00:02:10]I get a -3x is going to equal to a -16.

[00:02:14]Divide by -3, divide by -3. Now again, it's important. Negative divided by negative is going to be a positive, and 3 doesn't evenly divide into 16. And a lot of students at this point would say, "Uh-oh, I made a mistake." And sometimes that might be valid. Go back over and check your work. Did I actually make a mistake? Added 4, yes, that gives you -16. Yeah. Well, guess what?

[00:02:33]We are going to have a fractional answer. And guess what, ladies and gentlemen? That is a okay, okay? All right. Now, let's work into some fractions. We know we all love fractions. But first part, I'm not going to worry about the fraction over here.

[00:02:45]I'm just going to add a 1 to both sides, right? Cuz I always want to undo addition and subtraction first. So, now I have a 1/4x is equal to a positive 3.

[00:02:54]Now again, this is 1/4 * x, okay? So, you think about it like this. You can put parentheses in there. Now, you could divide by a 1/4 on both sides, but I'm pretty sure most students are like, "I just want to get rid of this fraction."

[00:03:05]So, what's another way that we can divide by a fraction, which is going to give us the same result? And that is simply going to be, yeah, dividing by a fraction is actually the same thing as multiplying by its reciprocal. So, I can multiply by a 4 over 1, which is the same operation as dividing by 1/4. But it makes the math a little bit easier, especially when you're dealing with fractions.

[00:03:25]Notice how this 4 over 1 * 1 over 4, that's just going to go to 1, right? So, there we have our coefficient is isolated, and then that's going to equal to a 12. Now, here is where a lot of students make their mistakes. A lot of students make mistakes here. And so, they see undo addition and subtraction.

[00:03:40]They see a subtraction, and they assume maybe like the 3 is a negative, or they add something over to the other side. I don't know. The best way that I can explain for students to work on a problem like this when you have your variable is in front or is behind your constant is just to rewrite it. And I would rewrite it like this.

[00:03:58]Not a -3 * x / 5, but a -3/5 x. And then notice this 3, it doesn't have a sign in front of it. So, it's a positive 3 is equal to 6. Now again, before I worry about the fractions, I'm going to subtract a 3 on both sides.

[00:04:14]So, we subtract a 3 and we have a -3/5 x. I don't know why I use parentheses.

[00:04:19]It is equal to a positive 3. Now, it's important this negative could go in the top, it could go in the bottom, right?

[00:04:24]Or it can go in front. Just do not put it in the top as well as in the bottom.

[00:04:27]Now again, notice here 3 * x / 5 is the same thing as a 3/5 * x. I want to write it like that because I want to understand just one operation. And again, just like we did over here, rather than dividing by 3/5, I'm just going to multiply by the reciprocal, which is going to be a -5/3.

[00:04:45]So, -5/3.

[00:04:47]Cool thing over here, 3 in the numerator, 3 in the denominator, those go ahead and divide out. And then x is going to equal to a just a -5 cuz a negative times a negative makes that a positive. So, then we're good to go over here. Now, let's get into the three-step. And the one thing I want you to understand about a three-step is because sometimes this gets grouped grouped together in a textbook with like two-step equations or it gets grouped with the multi-step. The one thing I want you to know about this is to get rid of your grouping symbol. There are two There is a grouping symbol in each of these problems. In this first example, the grouping symbol is going to be the parentheses. And then in the second example, the grouping symbol is going to be this division bar. Okay?

[00:05:23]Now, we use these and we understand these when we're using the order of operations. But solving linear equations, we're applying the reverse order of operations. So, there's two ways that we could get rid of the parentheses. We can apply the operation that is being applied to them, like we could distribute a -13 to both of these, and we could divide the two into both of these expressions.

[00:05:43]I think you could probably agree with me. I don't really want to multiply all of this through and then get a two-step equation, but you could, okay? But divide that through.

[00:05:52]And then over here, I don't want to divide a two into a 3x and a two into a three cuz even though two divides into six, which gives you three, two doesn't evenly divide into three halves and then I deal with another fraction and I just did fractions and even though I understand fractions, I don't want to work with fractions. So what else could we do? Well, instead of applying the operation, we can also undo the operation that's being applied to your grouping symbol, right? This inside this grouping symbol is being multiplied by -13. So why don't we just divide by -13 on both sides, right? I know I said undo addition and subtraction first, but that undo addition and subtraction is inside of the grouping symbol. So let's undo the grouping symbol so then we can apply those reverse order or then we can apply that two-step process.

[00:06:34]So when I divide by -13 on both sides, I get a 4x + 10 is equal to a positive 13.

[00:06:40]Now, you can see, ladies and gentlemen, oh, sorry, yeah, positive 13, yeah. Now I can subtract the 10 on both sides, I get a 4x is equal to a positive three.

[00:06:49]And then I just divide by four on both sides, x is equal to a 3/4. And then over here, again, to undo dividing by two, what I'm simply going to do is just multiply by two and I'm going to do that on both uh if that two is over that 15, that's a 15. Sorry, I didn't recognize I didn't have enough room.

[00:07:10]Here, I'll put it down here.

[00:07:12]There we go.

[00:07:14]Make myself a little bit of extra room.

[00:07:15]Where are you?

[00:07:17]Okay. Do do do do Sorry about that. All right.

[00:07:21]So I multiply by a two on both sides.

[00:07:23]Now those are going to divide out, that's going to give me a -30 so I get a 3x + 6 equals a -30. And then when I subtract a six on both sides, you owe me $30, you borrow six more, that's a -36.

[00:07:36]So 3x is equal to a -36, divide by three, divide by three, x is going to equal to a negative 12. And there you go, ladies and gentlemen.

[00:07:45]Hopefully now you have comfortability with being able to solve some linear equations. If you want to see some more examples or some practice, check out the next video I have for you here or the description down below. Cheers.