 To find the slope of a line between two points, we find the left-to-right run between the two points, find the rise, and then find the quotient, rise over run. But what if we don't have the two points? For example, suppose we want to find the slope of the line 3x plus 8y equals 24. Since we don't have two points on the graph of the line, we find two points. We'll graph the line. So if x equals 0, then equals means replaceable, and so our equation becomes, and we can solve for y to find. So y equals 3, and so 0, 3 is on the graph. Or if y is equal to 0, then our equation becomes, and we can solve for x to get, and so 8, 0 is a point on the graph. And now that we have two points on the graph, we can graph the line. And between the two points, we see that we have a run of 8 and a rise of negative 3, negative, since our height decreased, and the slope, rise over run, negative 3 eighths. Now part of the value of algebra is that we don't have to work with numbers, we can work a little bit more abstractly. So let's find the slope of the line y equals mx plus b. Well let's find two points. If x equals 0, then we find y equals b, and so this is the point x equals 0, y equals b. Now since this point is on the y-axis, we'll call it the y-intercept, and this allows us to conclude the line with equation y equals mx plus b has y-intercept 0, b. It's important to remember that the intercept is a point, and so we should always specify its coordinates as an ordered pair 0, b. But far more importantly, remember, don't memorize formulas, understand concepts. The reason we know where the y-intercept is, is we found it by letting x equals 0. To find a second point, we'll try x equal to 1. If x equals 1, we find that y is equal to, and so 1m plus b is on the graph, and the slope between our two points, 0, b, and 1m plus b, will be, which gives us another important result, the slope of the line with equation y equals mx plus b is m. And again, don't memorize formulas, understand concepts. We got this slope by finding two points on the graph and calculating the slope between them. So let's find the slope of the line with equation y equals 3x minus 7, and let's also find the y-intercept. So our theorems say that if the line has equation y equals mx plus b, we can read off the slope and the y-intercept. And so here, the line with equation y equals 3x minus 7, the slope is the coefficient of x. So our slope is 3, and our y-intercept has the form 0, b, and since b here is negative 7, our y-intercept is 0, negative 7. And in a kind and gentle universe, we'd always be given equations in the form y equals mx plus b. We don't live in that universe. The problem with formulas is they're a lot like all those internet apps. You have to abide by all the little rules that regard their use. So let's say we want to find the slope and y-intercept of the line 3x plus 8y equals 24. Well, we have a theorem that tells us how to find the slope and y-intercept if our line has the equation y equals mx plus b. And so the theorem requires the equation of the line to be in the form y equals mx plus b, so we need to solve for y. We can do that. So solving for y gives us... And so now we can use a theorem. The slope is the coefficient of x, negative 3, 8, and the y-intercept is 0, 3. Now, you might compare that to the solution we got earlier. So it's important not to rely too much on the formula because sometimes that will make you do a lot of extra work.