 Solving equations is one of the more important uses of mathematics. An important idea in math and in life is that how you speak influences how you think. And this is why it's so very important to get the terminology correct, because if you start calling things the wrong name, you'll start doing the wrong things with them. So what's an equation? An equation consists of two expressions linked by an equal sign. So for example, 4x plus 7 is an expression. 3y plus 8 is an expression. When I link the two expressions with an equal sign, 4x plus 7 equals 3y plus 8, we get an equation. What's important to remember is that an equation might be true or false. So here I have two expressions, 7 plus 3 and 10. 10, they are linked by this equal sign, and I have the equation 7 plus 3 equals 10, and this happens to be true. On the other hand, I have two expressions, 7 plus 3 and 73. I'll link them with an equals to form an equation, but this equation happens to be true. I mean false. And this brings us to an important idea. Solving an equation means finding values for the variables that will make the equation true. Now before we worry about actually solving equations, let's see what this means. So for example, does x equal 1 solve the equation x cubed plus 4x squared minus 12x plus 17 equals 25? And so what this means is we want to check to see if x equals 1 makes the equation true. So here's an important idea. Anytime we see an equal sign in mathematics, what we're claiming is that we can substitute one thing for the other. So this x equals 1 says that any place we see 1, we can replace it with x. Somewhat more usefully, any place we see x, we can replace it with 1. So I see x's, and we'll replace those x's with 1. Now at this point we need to introduce a little bit of new notation. An important idea to keep in mind in math and in life is that if you write something down, you're making a commitment. So as it's written, I'm making the commitment that 1 cubed plus 4 times 1 squared minus 12 times 1 plus 17 guaranteed equal to 25. But remember the whole point of this is to check to see if x equals 1 makes the equation true. I don't want to commit to saying left-hand side and right-hand side are the same thing. And so we'll indicate that by putting a question mark above the equals. And so now we can evaluate the arithmetic expression over on the left-hand side, and we end with this statement 10 possibly equal to 25. Now unless you're a politician who can't tell the difference between large and small, 10 and 25 are not equal. So x equals 1 does not make the equation true, so x equals 1 is not a solution. Or let's consider another example. Does x equals 2, y equals 3 solve the equation? 3x plus 2y equals 12. So again we'll replace x with 2 and y with 3 and see if we get a true statement. Substituting those in and doing some arithmetic. Since 6 plus 6 is actually equal to 12, then it follows that x equals 2, y equals 3 is a solution. How about an equation like x cubed plus 7x equals 22? Well again keep in mind that what we want to do when we solve is we want to find a value of x that makes the equation true. Now if we happen to be walking along the street and a value of x that makes this equation true falls out of a window and hits us on the head, well maybe we're walking in the wrong part of town because that would hurt. But if we should happen to find such a value of x then we've solved the equation. So one approach is to try what's called guess and check. We'll guess a value of x, if it works, if it makes the equation true, great we're done. But if the value doesn't work we try another value. And an idea that's very important to keep in mind as you move forward in your mathematics career is that we can always try guess and check. It may not be the best way of solving a problem but we can always start with it and see where it takes us. So let's try out different values of x. Now because we're guessing values of x it means we can pick any value that we want so let's try an easy value. How about let's try x equals 0 for starters. So every time I see x in my equation I'll replace it with 0 and see if we get a true statement. But we don't. And here's a useful idea in math and in life if it's not written down it didn't happen. So even though we tried x equals 0 and found it was not a solution we don't want to just make all this work disappear. I said we don't want to make the work disappear but it back. What we should do is we could record the fact that we tried x equals 0 but it was not a solution. So we'll write that down. And the reason that's useful is that we want to have a record of what we've tried so we don't try it again. Well that didn't work so let's try a different value we'll try x equals 1. And we find that x equals 1 is not a solution but we are closer to 22 so maybe we're going in the right direction. And this is an important issue for later on. For now we'll just record the fact that x equals 1 is not a solution and try again. Well let's try x equals 2. So replacing x with 2 every place we see it and 22 is actually equal to 22 and so we found a solution. Now if you think about this in some sense we got lucky in that our solution was found by our third guess. But maybe we're unlucky maybe our guesses have to be a lot more complicated. So the question that we want to ask is how much guess and check do we want to do? And the easy answer to that is as little as possible. Sometimes we have no way of avoiding it but we'd prefer it if there was another way of solving equations. And so this leads to the important idea to solve an equation for x or whatever the variable is is to write the equation in the form x equals stuff where stuff does not contain the variable x. So if our equation is x plus 3 equals to 5 this is not in the form x equals stuff so this is not solved for x. What about the equation x equals 5 over 2 plus x? Well this is in the form x equals stuff but we need to make sure that the stuff does not contain the variable x. So this equation is still not solved for x. How about x equals 7 minus 5 over 3? So here we have the equation in the form x equals stuff and what's over on the right hand side doesn't include the variable x. And so this is solved for x. And if x equals the cube root of pi cube plus 8y to power 5 the stuff doesn't include the variable x and so this equation is solved for x. So that's our destination. We'd like to write our equation in the form x equals stuff and the question is how do we get there? We'll take a look at that next time.