Slope-Intercept Equation: Exploring Lines on the Cartesian Plane

Cartesian Plane: Slope-Intercept Formula for Lines We will now derive the equation of a line using the point slope formula. The line we are working with passes through the point (2,1) and has a slope of M=1. Therefore, the point slope formula of the line is y-1= 1(x-2). Any point (x,y) that wants to be on this line must satisfy this equation. We can simplify this equation into the slope intercept formula: y = mx+b where m is the slope and b represents our y-intercept. Although the point-slope formula is widely used, it's worth noting that one of its key concepts is closely related to a different formula—the slope-intercept formula. First, however, we should take a moment to establish terminology. The line shown here has infinitely many points: (2,1) is 1, (3,2) is 1, (4,3) another. However, one point is really important for some reason: this one here—we call it the Y intercept.The Y intercept is the unique point where this line meets the y-axis. The X coordinate of this point is zero. The Y-axis consists of all points (x,y) with X coordinate zero. If we don't know something but we want to compute it and do algebra with it, we often give it a symbol, so that point is called (0,b), where b represents the Y intercept. If we know the equation for the line and a point on that line, then we can find what b must be. For example, if you know that point (0,b) is on the line L and you know the equation for L, then you can find out what b is. We know that (0,b) is on the line and so, if we plug in zero for X and b for Y, we have to make the truth. So b-1=1(0-2). Now we are going to take the slope intercept form of y = mx + b and apply it to the linear equation with slope m = -1 and y-intercept (0-1). Let's first rewrite it in slope intercept form with the point-slope formula, y-(-1)=m(x-0), where m = -1 and (0-1) is on the line. We now have a second equation for this line: y+1=m(x-0) or y=mx-1. This is the slope intercept formula for a line, written in point intercept form. Let us assume that a line has a slope of M and intersects the Y-axis at (0, b). Then, Y = Mx + b is an equation for the line. The slope is M and b is the Y intercept or the value of Y when x = 0. That process is often a nice, quick way of describing Y because it allows you to draw a line just by seeing it. For example, if we have the equation y=2x+1, let's draw that here. We know the y intercept is one. It's about right here. We know the slope is two which means steeper than 45-degree angle so about like that. Let's suppose that someone tells you to draw a line with the same Y intercept but sloped less positively. Then that could be, say, this one. Now let's suppose that someone else tells you to draw a line with the same slope as L (the purple line) but with an X intercept of negative one. Let's say down here. Now try your best to make it parallel to Y (the pink line). The slope tells you in this formula Y = Mx + b how to angle the line; the Y intercept tells you where to anchor it on the Y axis. So that's somehow much more pleasing than saying "point slope formula" every time someone asks you for the equation of a line. Let's finish with this example. The line L has two points (1,1) and (3,0) on it. Find an equation for L. Problem - L has points (1,1) and (3,0) on it. Find an equation for L. First step is let's draw point to C. So here is (1,1), here is about (3,0), and there is the line between them. All right, now let's find the equation for the line. We can do it point slope formula, we can do point intercept, whatever we want. First let's figure out the slope. So the slope of L is a line segment between (1,1) and (3,0). So M is equal to zero minus one divided by three minus one is negative 1/2. So this line has slope -1/2. And now to the point slope formula for the line. So let's take (1,1) as our point. We can create one equation for this line by using the formula y=mx+b. So we get y-1=-1/2(x-1). To find the Y intercept, we could use any number or other method. Here's a fun idea: We could also try the point (3,0) and then use the point-slope formula to find the equation of that line. Another equation for this same line is y=-1/2x-3. This may look like a contradiction because these two equations look very different; however, they are actually the same line. We can manipulate one of these equations into looking like the other by doing a little algebraic manipulation.