## Basic Equation Solving

### Transcript

In the previous set of videos, we're talking about algebraic expressions. And of course, the point there was to change to equivalent expressions that would always be true for all values of x. Now we're starting basic equation solving. And because weve learned a few important rules for changing algebraic expressions to equivalent expressions, this will help us in basic equation solving.

So this is a very introductory video. If youre good at algebra, you probably dont need to watch this video. In solving equations, the variable has either a single unknown value or a couple unknown values and the point is to find what the variable equals. So this is a very different game from the game of simply rearranging algebraic expressions.

Here we're actually trying to find the value for the variable. We need to solve for x, which means isolating x, that is, getting it by itself on one side of the equation. The basic mathematical constraint is that whatever we do to one side of the equation, we have to do to the other. In other words, we can add the same thing to both side.

We can subtract the same thing both from both sides, we can multiply or divide. Whatever we do to one side, whatever mathematical operation, we have to do that same mathematical operation to the other side. Mathematically, we are allowed to do many things, but strategically, there are very specific steps in solving any problem. These steps involved undoing the Order of Operations.

So it's very important to have this distinction of what is mathematically allowed, that's a broad field of possibility versus strategically what are the best steps to solve the problem. So suppose we have to solve this simple equation, 3x + 5 = 17. Theoretically, it should not be a big challenge to solve an equation like this but let's talk about this.

First of all, mathematically, there are many things we could do. For example, it would be mathematically correct to say to add 40 to both sides of the equation. Now there would be no reason to do that. But there being nothing in the laws of mathematics saying that that was a wrong thing to do, would not be mathematically wrong, it would simply be strategically wrong.

Just because we could do something, does not mean it is most strategic thing to do. There are many more things that we could do than are strategic to do, very important to keep that in mind. So what would be strategic? Think about the Order of Operations acting on x. First, we multiply the x by 3 then we add 5 to the product.

That's the order we need to undo. We begin by undoing the plus 5, by subtraction 5. What we do to one side, we must do to the other. So we subtract 5 from both sides of the equation. 3x + 5- 5, the 5 is cancelled on the left side. And on the right side, 17- 5 is 12.

Now we have to undo the multiply by 3, so we divide by 3. And of course, when we divide both sides by 3, we get x = 4, that's the answer. Now here's the same equation. Notice that we could begin by dividing both sides by 3. That would be mathematically allowed. Strategically, that would not be the smartest move. You see, if we divide it by three, every single piece, the 3x, the 5, and the 17, all would get divided by 3.

And so we'd wind up with this fraction equation. That would not be the best thing. So it's really good to subtract before we divide. Now why is that? Strategically, it's always important to think about the Order of Operations acting on x, and follow those steps backwards.

So what's going on here, why backwards? Think about it this way. When you put on footwear, the order is you put on your socks, then your shoes. Of course, when you remove footwear, you don't remove items in the same order. You have to take off the shoes first, then the socks. Undoing changes the order, both in footwear, and in many other articles of clothing, and in mathematical operations.

So it's just like shoes and socks, whatever goes on first comes off last. So here are a few practice problems. Pause the video and try these on your own. And here are the answers. And of course in each case, what we did was, we added or subtracted, got rid of the constant on the same side as the x before we divided.

What if x appears on both sides of the equation? Suppose we have something like this. Here, our initial strategy is to get all the terms involving x on one side, and all the constants on the other side. So here, we're gonna begin by adding 3x to both sides. So on the right side, the x we will just cancel.

And on the left side, we get 2x plus 3x equals 5x. Now that we have all the x's on one side, now we can just add 7 to both sides. We get 5x = 23 and divide. We can express the answer either as a improper fraction or a mixed numeral or a decimal depending on what the answer was looking for.

Here's some more practice problems. Pause the video and try these. And here are the answers. Everything we have covered in this module has been linear equations, that is, equations in which the highest power of x is 1. The rules become very different when we're no longer dealing with linear equation.

But everything I've said here is true for linear equations. So in solving equations, our goal is to find the unknown value of the variable. Mathematically, it's always legal to do any arithmetic operation to both sides of the equation. Strategically, we undo the Order of Operations to isolate the variable. And if the variable appears on both sides, we begin by collecting all terms with the variable on one side.

Read full transcriptSo this is a very introductory video. If youre good at algebra, you probably dont need to watch this video. In solving equations, the variable has either a single unknown value or a couple unknown values and the point is to find what the variable equals. So this is a very different game from the game of simply rearranging algebraic expressions.

Here we're actually trying to find the value for the variable. We need to solve for x, which means isolating x, that is, getting it by itself on one side of the equation. The basic mathematical constraint is that whatever we do to one side of the equation, we have to do to the other. In other words, we can add the same thing to both side.

We can subtract the same thing both from both sides, we can multiply or divide. Whatever we do to one side, whatever mathematical operation, we have to do that same mathematical operation to the other side. Mathematically, we are allowed to do many things, but strategically, there are very specific steps in solving any problem. These steps involved undoing the Order of Operations.

So it's very important to have this distinction of what is mathematically allowed, that's a broad field of possibility versus strategically what are the best steps to solve the problem. So suppose we have to solve this simple equation, 3x + 5 = 17. Theoretically, it should not be a big challenge to solve an equation like this but let's talk about this.

First of all, mathematically, there are many things we could do. For example, it would be mathematically correct to say to add 40 to both sides of the equation. Now there would be no reason to do that. But there being nothing in the laws of mathematics saying that that was a wrong thing to do, would not be mathematically wrong, it would simply be strategically wrong.

Just because we could do something, does not mean it is most strategic thing to do. There are many more things that we could do than are strategic to do, very important to keep that in mind. So what would be strategic? Think about the Order of Operations acting on x. First, we multiply the x by 3 then we add 5 to the product.

That's the order we need to undo. We begin by undoing the plus 5, by subtraction 5. What we do to one side, we must do to the other. So we subtract 5 from both sides of the equation. 3x + 5- 5, the 5 is cancelled on the left side. And on the right side, 17- 5 is 12.

Now we have to undo the multiply by 3, so we divide by 3. And of course, when we divide both sides by 3, we get x = 4, that's the answer. Now here's the same equation. Notice that we could begin by dividing both sides by 3. That would be mathematically allowed. Strategically, that would not be the smartest move. You see, if we divide it by three, every single piece, the 3x, the 5, and the 17, all would get divided by 3.

And so we'd wind up with this fraction equation. That would not be the best thing. So it's really good to subtract before we divide. Now why is that? Strategically, it's always important to think about the Order of Operations acting on x, and follow those steps backwards.

So what's going on here, why backwards? Think about it this way. When you put on footwear, the order is you put on your socks, then your shoes. Of course, when you remove footwear, you don't remove items in the same order. You have to take off the shoes first, then the socks. Undoing changes the order, both in footwear, and in many other articles of clothing, and in mathematical operations.

So it's just like shoes and socks, whatever goes on first comes off last. So here are a few practice problems. Pause the video and try these on your own. And here are the answers. And of course in each case, what we did was, we added or subtracted, got rid of the constant on the same side as the x before we divided.

What if x appears on both sides of the equation? Suppose we have something like this. Here, our initial strategy is to get all the terms involving x on one side, and all the constants on the other side. So here, we're gonna begin by adding 3x to both sides. So on the right side, the x we will just cancel.

And on the left side, we get 2x plus 3x equals 5x. Now that we have all the x's on one side, now we can just add 7 to both sides. We get 5x = 23 and divide. We can express the answer either as a improper fraction or a mixed numeral or a decimal depending on what the answer was looking for.

Here's some more practice problems. Pause the video and try these. And here are the answers. Everything we have covered in this module has been linear equations, that is, equations in which the highest power of x is 1. The rules become very different when we're no longer dealing with linear equation.

But everything I've said here is true for linear equations. So in solving equations, our goal is to find the unknown value of the variable. Mathematically, it's always legal to do any arithmetic operation to both sides of the equation. Strategically, we undo the Order of Operations to isolate the variable. And if the variable appears on both sides, we begin by collecting all terms with the variable on one side.