 We've been graphing in rectangular coordinates, but because I was trained the way that I was, I might slip from time to time and talk about Cartesian coordinates or Cartesian graphing. Well, where does that come from? A little explanation is in order. In 1637 the French mathematician Pierre de Fermat expressed a very important idea. Every algebraic equation in two variables corresponds to some geometric curve. From the same time, Rene Descartes expressed another idea. Every geometric curve can be described by an algebraic equation. Now what we're actually doing has more in common with what Fermat was describing than with what Descartes was describing. And so we should speak of Fermatian coordinates. But because Descartes was a famous philosopher, and Fermat was just a lawyer, we instead speak of Cartesian coordinates. So let's formalize Fermat's concept. The graph of an equation in x and y consists of all points hk, where x equals h, y equals k satisfies the equation. And what this means is that if x equals h, y equals k satisfies the equation, then the point hk is on the graph. If x equals h, y equals k does not satisfy the equation, the point hk is, wait for it, not on the graph. So for example, let's try to determine which of the points 3, 5, 2, negative 3, negative 3, 1 is on the graph of 5x minus 2y equals 16. So remember a point is on the graph of an equation if the x and y values make the equation true. So the point 3, 5 has x equal to 3, y equal to 5, and we'll check to see if these values make the equation true. Equals means replaceable, so we'll replace x with 3 and y with 5 and do a little arithmetic. And since 5 is not equal to 16, the equation is false, and so the point 3, 5 is not on the graph of 5x minus 2y equals 16. What about the point 2, negative 3? Well this point has x equal to 2, y equal to negative 3, and so we check. Equals means replaceable, so we'll replace, we'll do a little arithmetic, and this is true. So the point 2, negative 3 is on the graph of the equation. So the point negative 3, 1 has x equal to 3, y equal to 1, so we can check, and alternative facts notwithstanding, this statement is false. And so we know that the point negative 3, 1 is not on the graph. We're going to see if a point is on the graph is relatively easy. It's a little bit harder to find points on the graph, so if I want to find 2 points on the graph, I need to find a pair of values x equals h, y equals k, that satisfy the equation. And it's useful to keep in mind, we already know how to solve equations in one variable, so let's transform this into an equation in one variable by choosing a value for the other. So maybe we'll pick a value for x, if x equals 1, equals means replaceable, so we'll replace x with 1 in our equation. And we can try to solve this equation, but this looks hard to solve. Since we can choose any value of x or y we want, let's choose a different value. Well we picked a value for x, let's pick a value for y. How about if y equals, oh I don't know, I like the fraction 5 17s, equals means replaceable, so replacing y with 5 17s in our equation gives us. And we can solve this equation for x, but this looks hard to solve too. So let's try something else. If x equals 0, equals means replaceable, so replacing x with 0 in our equation gives us. And we can simplify this a little bit. And this equation looks a lot easier to solve. We can divide both sides by 5 and get y equal to 6. And so since x equals 0, y equal to 6 gives us a true statement, then the point with x equal to 0, y equal to 6 is on the graph. And so we can say that the point 0, 6 is on the graph. Now we could pick a different value for x, but we already tried that and we didn't like the equation that we got. So let's try y equal to 0. If y equals 0, equals means replaceable, so we'll replace y with 0 everywhere we see it. And the resulting equation is easy to solve, we'll solve this for x and get. And since y equals 0, x equals 10 gives us a true statement, then the point with y equals 0, x equals 10 is on the graph. Now remember when we give the coordinates of a point, the x coordinate is given first. And so that means the point 10, 0 is on the graph. How about something like this? Now since this has the form of a formula for y in terms of x, we might just pick values of x and compute. So if we let x equals 0, substituting that in, we solve for y. So x equals 0, y equals 7, 0, 7 is a point on the graph. If we let x equals 1, substituting that in, we find that y is equal to 7 and 2-fifths. So x equals 1, y equals 7 and 2-fifths is also a point on the graph. So I know that most of you out there are wanting to deal with fractions and you're really dying to pick a value of x like 3-7s or something like that. We might want to make this a little bit easier. And a useful thing to keep in mind is that if you multiply a fraction by its denominator, the fraction becomes an integer. And so that suggests we might want to choose x equal to 5, the denominator of our fraction. If we let x equals 5, then substituting that into our formula for y gives us 9. And so the point on our graph, x equals 5, y equals 9, that's the point 5, 9.