Point-Slope Form and Slope-Intercept Form
Slope Intercept Form: y = mx + b, where m is the slope and b is the y-intercept. Point-Slope Form: y-y_1=m(x-x_1), where m is the slope and (x_1,y_1) is a point on the line.
Slope-Intercept and Point-Slope Forms
Typically, when you’re dealing with a problem that requires graphing of linear equations, you’re going to want to get your equations into a form called slope-intercept form. I talked briefly about this in a previous video, but I wanted to define it a little bit more clearly here.
Slope-intercept form looks like this. You have y on the left side, and on the right you have your slope, usually represented by m times x, your variable x, plus b (your y intercept). The slope, as I mentioned, is defined as the change in the vertical coordinate divided by the change in horizontal coordinate over any given interval on the line.
B, the y intercept, is the point on the y axis where the line crosses. You can either use these two terms to plot your line or you can identify these two terms given the line by looking at and analyzing where the line crosses the y axis and what the slope is. This is slope-intercept form.
This is the form that you want to get your equations in if you’re having to graph them. Frequently, what you may have to do is convert from another form, which is just going to involve solving the equation for y in terms of x. That will generally get you the slope-intercept form.
Another form that you may have to use in some instances is called the point-slope form. The point-slope for is useful if you’re given a point that a line goes through and the slope of a line and you’re asked to find the slope-intercept form of that line equation. The point-slope form of a line looks like this.
You have your y variable minus the y coordinate of the point you’re given. We’ll represent it with y1 is equal to the slope times your x variable minus x coordinate of the point you’re given (x1). Y minus y1 equals m times x minus x1. Let’s take an example of how we can get from the point-slope form to the slope-intercept form.
Let’s say we’re asked to find the slope intercept form of the equation of the line that passes through point 5,12 and has a slope of 3. This is going to be our point. X1 equals 5, y1 equals 12, and 3 is our slope. That’s going to be m.
Now we can plug in these values into the equation, and we can rearrange it to get our slope-intercept form. Y minus y1 (12) is equal to 3 times x minus x1 (5). Y minus 12 equals 3 times x minus 5. We need to distribute this 3. This will give us 3x minus 15. Then, we need to isolate y.
We’ll add 12 to both sides of the equation, and that gives us y equals 3x minus 15 plus 12 is going to be minus 3. Now we have the slope-intercept form of the equation of a line that passes through this point with this slope. That’s one example of using the point-slope form to solve a problem.
Most frequently, this is the form of a line, or the form of the linear equation that you want to have when you’re dealing with graphing.