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Converting Between Forms

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Today we're going to talk about converting between different forms of linear equations. So we're going to start by looking at three different forms of linear equations, slope intercept form, slope point form, and standard form. And then we'll do some examples converting between these three forms.

So let's start by reviewing three different forms of linear equations. The first form is slope intercept form. These equations are in the form y equals mx plus b. And in this equation, the value that we have for m tells us the slope of the line and the value that we have for b tells us the y-intercept for our line. So in slope intercept form, we can easily see, just by looking at the equation, what the slope of the line is and what the y-intercept of the line is.

The second form is called slope point form. Equations in this form look like this-- y minus y1 is equal to m times x minus x1. And in equations in this form, the value for m is going to tell us the slope of our line. And our values for x1 and y1 will tell us a point on that line. So equations written in this form, just by looking at the equation, we can easily see what the slope of the line is and we can see that the x and y-coordinates of a point on that line.

And then the last form is called standard form. Equations written in standard form look like ax plus by is equal to c. And when we're looking at equations in standard form, we cannot directly see what the slope, the y-intercept, or a point on that line is. So when you are solving problems involving equations of lines, you may want to convert between any of these forms depending on the information that you have and the information that you're trying to find. So let's see how we can convert between these different forms of equations

So from my first example, I have an equation that is written in standard form. And I want to write it or convert it into an equation that is written in slope intercept form. So remember that when an equation is written in standard form, I cannot directly see what the slope of the line, the y-intercept of the line, or a point on that line just by looking at the equation. But by converting it into slope intercept form, I can just look at the equation and see what the slope is and what the y-intercept is.

So when I want to write this equation in slope intercept form, I see that I want to isolate my y variable on one side of the equation. So to do that, I'm going to start by canceling out this 3x term. I'm going to do that by subtracting 3x from both sides. Here this will cancel. And I'll be left with 5y. And on the other side, I'm left with 10 minus 3x.

Next I'm going to cancel out of this 5 by dividing both sides of the equation by 5. And now I've isolated my y variable. So I have y is equal to. And on this side, I'm going to split up this fraction to be 10 over 5 minus 3x over 5.

And now I can simplify this by doing a couple of things. First I'm going to simplify 10 over 5 to be just 2. So y is equal to 2 minus 3x over 5. And second, to make this look exactly like slope intercept form, I want to flip these two terms so that the term with the x in it comes first and then the term, the constant term, comes last at the end.

So my equation is now going to become y is equal to negative 3 over 5x plus 2. So now I can just look at my equation. Because it's written in slope intercept form, I can see that my slope is going to be negative 3 over 5. And my y-intercept is going to be positive 2.

So for my second example, I've got an equation of a line that is written in slope point form. And I'll want to write it as an equation written in slope intercept form. So this equation is written in slope point form. I can see that it has a slope of 3. This is my value for m. And I can also see that a point on the line is going to have an x value, an x-coordinate of 1 and a y-coordinate of 2. So the point 1, 2 is a point on this line.

So let's go ahead and convert this equation into slope intercept form. And then we'll be able to also see the y-intercept of this line. So again, to solve this or to convert it into slope intercept form, I want to isolate my y variable. So I'm going to start by adding 2 to both sides of my equation. Here this will cancel. And I'll just have y is equal to 3 times x minus 1 plus 2.

Simplifying this, I'm going to distribute my 3 to both terms on the inside of the parentheses. So this will give me y is equal to 3x minus 3. And then I'm going to bring down my plus 2 at the end.

And then finally, I can just combine these two terms, negative 3 and 2. And that's going to give me a negative 1. I'll bring down my other terms of my equation. And I'm left with y is equal to 3x minus 1. So again, I can see that the slope of the line is 3. But now I can also see that the y-intercept of this line is going to be negative 1.

So for my last example, I've got an equation that is written in slope intercept form. And I'm going to convert it to an equation that is in standard form. So an equation that's written in standard form has the constant, the value for c, by itself on one side of the equation. So I want to move any other terms besides the constant to the other side of the equation.

So this negative 3x I want to move to the other side of my equation. So I'm going to do that by adding 3x to both sides. Here this will cancel. And I can't add y and 3x because they have different variables, so they're not like terms. So this side of the equation is just going to become 3x plus y. And I'm left with 8 on the other side. So now this equation is written in standard form.

Now, one thing that is nice about standard form is that you can pretty easily identify what the x and y-intercepts are. And that's because, for example, if we want to find the x-intercept, we remember that at the x-intercept, y is just 0. So if y is going to be 0, then I just need to solve the equation 3x is equal to 8, again, because my y value is just 0. So it's no longer there. Dividing both sides by 3, I would have that my x-intercept is going to be equal to 8/3, or approximately 2.67.

Similarly, if I wanted to find my y-intercept, I know that my x value is going to be 0. So this whole term is just going to be 0. And I'll just have the equation y is equal to 8. So 8 is my y-intercept.

So let's go over our key points from today. Make sure you get them in your notes if you don't already so you can refer to them later. The slope and y-intercept can be determined easily by looking at the equation of a line written in slope intercept form. And the slope and a point on a line can be determined easily by looking at the equation of a line written in slope point form. And finally, the x and y-intercepts can be found easily by solving the equation of a line written in standard form for x and for y.

So I hope that these key points and examples helped you understand a little bit more about converting between forms of linear equations. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.

Formulas to Know

- Point-Slope Form of a Line
- Slope-Intercept Form of a Line
- Standard Form of a Line