 There's a saying that a picture is worth a thousand words now since this video is running by at 24 pictures per second That means you've just seen a novel Although since it's been the same picture the novels are pretty boring one But I digress let's talk about the mathematical equivalent of a picture the graph of a line Now remember that any point in the plane can be specified by its Coordinance x and y a graph consists of a set of coordinates For example, I might try to sketch the graph of the following set of coordinates 1 3 1 5 and 2 4 So this first set of coordinates 1 3 Specifies a point that has x coordinate 1 and y coordinate 3 So we'll plot that point, but we'll also label its coordinates a good habit to get into Next I'll sketch the point with x coordinate 1 y coordinate 5 and Finally the point with x coordinate 2 y coordinate 4 Now it's important to understand that because our graph is given as these three points only Then our graph consists of these three points only Sometimes we have too many points to list them conveniently and so instead we represent the graph by writing the Relation between the x and y values For example, I might say something like this Suppose the y values of a point on a graph are three times the x values First of all we want to express this as an equation and then find three points on the graph Since the y values are three times the x values Then whatever the x value is the y value will be three times that amount and we can express this as Y equals 3x Now to find three points on the graph what I want to do is I want to find three sets of x and y values Now I have this relationship y equals 3x since this is going to be true for any x value We can choose any x value that we want So let's choose How about x equals zero as an easy value to work with if x equals zero Then our formula tells us that y is equal to Zero and so the point zero zero is on the graph Well, let's pick another value for x. How about Let's choose x equals one If x equals one our formula tells us that y is equal to Three so the point one three is also on the graph And if I choose x equal to two our formula tells us that y is equal to six And so we know the point two six is on the graph Now while we know these three points are on the graph We don't know if there are any others on the graph and we could find more by choosing different values of x And if I were endlessly patient I could fill in the space between these points with many many many more points But for reasons that won't be clear until you take calculus We can connect the dots to form a continuous graph Mathematicians say that the equation y equals three x corresponds to the curve shown And it's important to recognize two things First mathematicians are not very good at coming up with new and exciting names for things This is a curve and there's no real need to call it anything else and when a mathematician talks about curves They include things like this straight line So for example, let's draw the curve corresponding to the equation y equals five So again the equation describes a relationship between the x and y values on the curve As before we can pick a value for x and find the value for y So I'll pick x equals three because I like the number three then our formula tells us that y equals five So three five is a point on the graph I'll pick another value for x if x equals ten then our formula tells us that y equals Five so ten five is a point on the graph and if y equals well Why not let's use this horrible mess then our formula tells us that y equals Five and so I have a point on the graph after plotting these points I can connect the dots and carefully label what I've done Which gives me the graph of the equation y equals five Now let's pause a moment and think about this process if we had y equals any other number We'd still have the same value of y regardless of the x value And so we'd still get a graph that looks very much like this and this leads to the following theorem The graph of y equals C where C is any constant is a horizontal line Let's try another one. How about let's try and graph x equals three So as before we'll try and find pairs of x and y values So if x equals two then our formula says x equals three but that's impossible because if x equals two x can't be three and What this says is we can't pick values for x So we should Pick values for y instead So if y equals one then formula says x is equal to three And so we know that the point three one is on the graph If y equals five then x equals Three so three five is a point on the graph and if y equals eight our formula says that x equals three So three eight is another point on the graph And if we connect the dots we'll get our continuous curve and again if x is equal to any other number we're going to get a very similar graph and So this leads to the result the graph of x equal to C where C is any constant is a vertical line And let's pause a moment and see what we've gotten if y equals C is a horizontal line and x equals C Is a vertical line then what happens when we have an equation that involves both x and y like 5x plus 3y equals 15 as before we'll find some points But before we proceed we'll throw down a useful definition and a theorem The standard equation of a line is ax plus b y equals C Where a b and C are real numbers Well, actually I'm putting Descartes before the course Because we should actually point out that the graph of ax plus by equals C is a straight line So let's take a look at the graph of 3x plus 5y equals 15 So our definition says that we have the standard equation for a line and The theorem says that when we graph this we will actually get a straight line So from geometry we know that we only need two points to graph a line So let's find two points that satisfy this equation Since we get to pick our x or y values We want to pick the hardest possible value so that people will marvel at our awesomeness Or we might actually want to pick the easiest values that we can work with So we like zero if x equals zero Then substituting that into our formula gives us an equation that we can solve for y and we find Y equals five so zero five is a point on the graph While we could pick a different value for x and solve for y We might find it a little bit easier to let y equals zero So we'll let y equals zero and substitute that value into our equation and then solve for x And if we do that we find that three zero is also a point on the graph And because we know that this graph will produce a straight line We'll draw the straight line running between these two points