 Solving equations is an important use of mathematics, and among all the types of equations that you could solve, solving linear equations is probably the most fundamental and the most important. So what's a linear equation? A linear equation is an equation that involves, wait for it, linear expressions. Well, that may not be particularly informative if you don't know what a linear expression is, so here goes, an expression where all terms have Degree 1 or 0 is a linear expression. For example, 3x plus 5y plus 17, all of our terms have Degree 1 or 0, this is a linear expression. 23, the only term here has Degree 0, so this is a linear expression. Square root 17x, our variable x has exponent 1, so this is a linear expression. You have to be a little bit careful with hidden exponents. 1 over x plus 7 is not a linear expression because 1 over x is x to power negative 1. Likewise, square root x plus 7 is not a linear expression because square root of x plus 7 is x plus 7 to power 1 half. If two linear expressions are related by an equal sign, we have a linear equation. And you might ask, who cares? And there are two answers to that. The one boring, dull, but practical answer is that if you can solve an equation at all, sooner or later you're going to have to solve a linear equation. So being able to solve linear equations is a prerequisite for being able to solve any other type of equation. But that's a boring practical answer, sort of like being told to eat your vegetables or to cut out the fried food because it's good for you. I mean, nobody ever says you need to eat more chocolate cake and double up on the French fries. So here's a different reason why linear equations are extremely important for mathematics and mathematicians. Every linear equation can be solved for any of its variables. And that's amazingly neat because it says that any linear equation, no matter how horrible it looks, can be solved for any of the variables. This is not true for other types of equations. So how can we do that? It's useful to think about the following analogy. Solving an equation is very much like unwrapping a package. And when you unwrap a package, the very first thing you do is you start by taking out the bubble wrap. Uh, maybe we don't do that. We always start with the outermost wrapping and work our way in. Solving an equation proceeds in exactly the same way. Now, what this means is we'll have to talk a little bit about what are called inverse operations to unwrap an arithmetic expression. We apply an inverse operation. And here's why it's so important to be able to identify the type of expression you have in front of you. Because remember, the type of expression is determined by the last operation performed. And if we think about this as unwrapping a package, that last operation is the first thing we're going to have to deal with. It's the outermost part of the package. So let's consider our two basic arithmetic operations, addition and subtraction. If our expression is a sum where we add something, the inverse operation is a subtraction where we'll subtract something. Likewise, if the expression is a difference where we're subtracting something, the inverse operation is going to be a sum where we'll add something. So let's consider this thing, x plus 7 equals 15, or since I have two expressions joined by inequality, this is actually an equation, and we'll try to solve it. Because the only variable here is x, it's implied that when we solve this, we actually want to solve it for x. So remember that what we mean by solving for x is we want to rewrite this equation in the form x equals stuff where the stuff does not contain the variable x. And so what we want to do is we want to get x by itself on one side of the equality with the stuff on the other side not including x. Well, the good news is we're halfway there. The right-hand side does not contain the variable x. So let's take a look at this. Over on the left-hand side, we have a sum. So we're adding 7. And so our inverse operation is going to be to subtract 7. So let's do that. And here's an important idea. While we could just subtract 7 from one side, if we do that, we no longer have an equality. And though in general, we'll find that inequalities are far, far, far, far, far more important than equalities, equalities are easier to work with. And so what we'd like to do is we'd like to maintain the equality all the way through. So since we want to maintain equality, we have to do the same thing to both sides. So if I'm subtracting 7 from the left, I'll subtract 7 from the right. Over on the left-hand side, the plus 7 minus 7 are inverse operations. So it's as if we did nothing. And so we just have x. Over on the right-hand side, we have 15 minus 7. And so we'll do that arithmetic operation. And we get 8 over on the right-hand side and a new equation, x is equal to 8. It's important to recognize that this is still an equation. This is still something equal to something else. But in this particular case, this equation is solved for x. Our equation is in the form x equals stuff, where stuff does not contain the variable x. And so our solution, x equal to 8. Now it's important to understand that this is a claim that if x is equal to 8, then this statement x plus 7 equals 15 is going to be true. Now we should always get in the habit of checking our solutions. So we'll take our original equation and then substitute in x equals 8 and see if we get a true statement. And since this is a true statement, then we can say that x equals 8 is a solution. How about another equation? x minus 3 is equal to the additive inverse of 8. So again, very helpful to identify what type of expression we have. x minus 3 is a difference. We're subtracting 3. So our inverse operation will be to add 3 and since we want to maintain equality, we'll have to add 3 to both sides. Over on the left, we have x minus 3 plus 3. Well, that'll just give us x. Over on the right-hand side, we have additive inverse of 8 plus 3 and that'll give us additive inverse of 5 or you might say negative 5. And again, we'll check our answer. If x equals negative 5, we'll substitute that into our original equation and see if we get a true statement. And we have a true statement, so x equals additive inverse of 5 is in fact a solution.