Cartesian Plane: Slope-Intercept Formula for Lines We will now derive the equation of a line using the point slope formula. The line weare 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 onthis line must satisfy this equation. We can simplify this equation into the slopeintercept 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 keyconcepts is closely related to a different formula—the slope-intercept formula. First,however, we should take a moment to establish terminology. The line shown herehas infinitely many points: (2,1) is 1, (3,2) is 1, (4,3) another. However, one point isreally important for some reason: this one here—we call it the Y intercept.The Yintercept 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 Xcoordinate zero. If we don't know something but we want to compute it and doalgebra with it, we often give it a symbol, so that point is called (0,b), where brepresents the Y intercept. If we know the equation for the line and a point on thatline, then we can find what b must be. For example, if you know that point (0,b) is onthe line L and you know the equation for L, then you can find out what b is. We knowthat (0,b) is on the line and so, if we plug in zero for X and b for Y, we have to makethe 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 thelinear equation with slope m = -1 and y-intercept (0-1). Let's first rewrite it in slopeintercept form with the point-slope formula, y-(-1)=m(x-0), where m = -1 and (0-1) ison 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 thevalue of Y when x = 0. That process is often a nice, quick way of describing Y because it allows you to drawa line just by seeing it. For example, if we have the equation y=2x+1, let's draw thathere. We know the y intercept is one. It's about right here. We know the slope is twowhich 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 butsloped less positively. Then that could be, say, this one. Now let's suppose thatsomeone else tells you to draw a line with the same slope as L (the purple line) butwith an X intercept of negative one. Let's say down here. Now try your best to makeit 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'ssomehow much more pleasing than saying "point slope formula" every timesomeone 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 anequation 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 isthe line between them. All right, now let's find the equation for the line. We can do itpoint slope formula, we can do point intercept, whatever we want. First let's figureout the slope. So the slope of L is a line segment between (1,1) and (3,0). So M isequal to zero minus one divided by three minus one is negative 1/2. So this line hasslope -1/2. And now to the point slope formula for the line. So let's take (1,1) as ourpoint. We can create one equation for this line by using the formula y=mx+b. So weget 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 theequation of that line. Another equation for this same line is y=-1/2x-3. This may looklike a contradiction because these two equations look very different; however, theyare actually the same line. We can manipulate one of these equations into lookinglike the other by doing a little algebraic manipulation.