Hi, this is Anthony Varela. And in this tutorial, we're going to be converting between forms of linear equations. So first, we're going to review what those different forms are. Then we're going to talk about why we might want to convert in the first place.
What are we going to gain or lose by rewriting a linear equation from one form to another? And then finally, we're going to go through some examples. And I'll show you how to convert between different forms.
So first, the forms of linear equations. Well, we have slope intercept form, y equals mx plus b. And the great thing about slope intercept form is that we can clearly see what the slope of the line is by that the value of m. And we can see what the y-intercept is using that value for b.
Point slope form is y minus y1 equals m times x minus x1. And you're going to prefer this form once again, if you'd like to easily identify the slope by that variable m. But you're also going to want to use this form if you need to find a point on the line because we can just look at x1 and y1. That forms a coordinate pair, which is a point on the line.
And then we have standard form ax plus by equals c. And the great thing about standard form is that we can pretty easily calculate the y-intercept, but also the x-intercept as well. So why would you want to convert between forms? Well, what info do you need?
Are you looking for a y-intercept? Are you looking for an x-intercept? Maybe you want to find the slope or a point in the line xy? That's what's going to help you make your decision about what form you'd like to convert into. What info do you need?
So let's go through some examples. Our first example has us working with an equation in standard form. And we'd like to convert it into slope intercept form. So right now, I don't know what the slope of the line is. I don't know what the intercept is.
I can find that by converting into slope intercept form. Well, notice that in slope intercept form, I have y by itself on one side of the equation, and everything else is on the other side. So I need to do that in this equation written in standard form.
So the first step I'm going to take then is to subtract 8x from both sides of this equation. So now I have just my y term, negative 3y. And I just like to write in a x term first. So I have negative 8x plus 5.
Well, I still have that negative 3 out in front of y that I'd like to cancel out. So I'm going to divide both sides by negative 3. So I have just y on one side of the equation. Then I have negative 8x divided by negative 3. So that's a positive 8/3 x. And then I have 5 divided by negative 3, so a negative 5/3.
So now I have it written in slope intercept form. So then I can match up my m value with 8/3 and my b value with negative 5/3. So my slope is 8/3, and the y-intercept is negative 5/3.
So in our next example, we're going to be taking an equation written in point slope form and writing it into slope intercept form. So looking at this equation right here, we can see that the slope of the line is 3. And we know that it passes through the points-- through the point 1, negative 2 because that's what we get from this point slope form, the slope and a point on the line.
But what we don't know from this equation is the y-intercept where it crosses the y-axis. So that's why we might want to convert into slope intercept form. So the first thing that I'm going to do is distribute that slope of 3. Multiply that by x and multiply that by negative 1.
So now I have y plus 2 equals 3x minus 3. It's looking very close to slope intercept form. I just need to now move this constant term that's attached to y over to the other side of the equation, isolate y. So I'm going to subtract 2 from both sides.
So now I have y equals 3x minus 5. So now this confirms that the slope of the line is 3. And the y-intercept occurs at x equals 0, y equals negative 5.
So our last example has us working with an equation in slope intercept form. But I'd like to convert it into standard form. So here I have y equals 2x plus 3. And I'd like it to look like ax plus by equals c.
So what I notice is that I have an x term and a y term on one side of the equation and then a constant number on the other side. So what I need to do then is move this 2x to the other side of the equation. So I'm going to subtract 2x from both sides.
So now I have negative 2x plus y equals 3. Now a picky thing about standard form is that we generally prefer that our a value is non-negative. So we want to make that a positive value. It's simple enough to do. All we need to do is multiply the entire equation by negative 1, or in other words, change the sign of everything.
So we go from negative 2x to positive 2x, from plus y to minus y. And then we go from positive 3 to negative 3. So here's my equation then in standard form, 2x minus y equals negative 3.
Now I told you that the great thing about standard form is that we can easily calculate the intercepts, both x and y. So thinking about the x-intercept, this is when y equals 0. And the y-intercept is when x equals 0. Now when y equals 0, our equation then just becomes 2x equals negative 3.
We don't need to write y if it's 0. So then we know then that the x-intercept is negative 3/2. So there's the coordinate then of our x-intercept, negative 3/2 comma 0. How about the y-intercept?
Well, when x equals 0, we don't need to write the entire x term, no matter what that coefficient is. We know just that negative y equals negative 3. So if negative y equals negative 3, positive y equals positive 3. So the coordinate then of my y-intercept is 0, 3.
So let's review converting between forms. Well, their slope intercept form, y equals mx plus b, point slope form, y minus y1 equals m times x minus x1. And then we have standard form, ax plus by equals c.
Now why would you want to convert between forms? Well, slope intercept form tells you what the slope and the y-intercept are. Point slope form tells you what the slope is. And it also gives you a point xy on the line. And standard form lets you easily calculate y-intercepts and x-intercepts.
So thanks for watching this video on converting between forms. Hope to see you next time.