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Forms of Linear Equations

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Today we're going to talk about forms of linear equations. So we're going to look at different ways that you can write the equation for a line. The three forms that we're going to look at are slope-intercept form, slope-point form, and standard form. And I'll give you some examples of equations written in each of these forms.

So our first form of a linear equation is called slope-intercept form. And in this form, the equation is going to look like y is equal to some number m times x plus another number b.

We call this slope-intercept form because in this form, we can directly see what the slope and the y-intercept of the line is because the value that we have for m is going to be the slope of our line, and the value that we have for b is going to be the y-intercept of our line.

So for example, if I have the equation y equals 1/2 x minus 3, this equation we can see is in slope intercept form. And so the slope of my line is going to be 1/2 and the y-intercept of my line is going to be negative 3.

So if I remember that the definition of a y-intercept is that it's the value of y when x is equal to 0, I can verify that the y-intercept of this line is going to be negative 3 by substituting 0 in for my x in the equation.

If I were to do that, I would have y is equal to 1/2 times 0 minus 3. 1/2 times 0 is just going to make this whole term into 0. And then I'll have minus 3. That's going to be equal to y. 0 minus 3 is just negative 3. So again, I can see that the value of y is going to be negative 3 when x is 0, which means that my y-intercept is negative 3.

Let's look at our next form of a linear equation. Our second form of a line is called slope-point form. And equations in this form look like this. y minus some value for y1 is equal to some value for m times x minus some value for x1.

And we call this slope-point form because from this equation we can see directly the value of our slope of the line, and also we can see the coordinate points-- the coordinate pair of a point on that line.

So in the equation, the value that we have for m is going to tell us the slope of the line. And the value that we have for x1 and y1 will tell us a point on that line.

So for example, if I have the equation y minus 8 is equal to 2/3 times x minus 9, I know that the slope of my line is going to be 2/3 because that's the value that I have for m. And I know that a point on the line is going to be 9, 8.

So if I'm looking at this equation, I know that my slope is 2/3 and that also a point on the line is going to be the point 9, 8.

So the last form of a linear equation we're going to look at is called standard form. And equations in standard form look like this. Some value for a times x plus some value for b times y is equal to some value for c.

So an example of that would be 8x plus 11y is equal to 24, where eight is our value for a, 11 is our value for b, and 24 is our value for c.

So when you're looking at equations written in standard form, you cannot directly see what the slope of the line is, what the y-intercept of the line is, or what a point on that line is in the way that you can see when you're looking at an equation written in slope-intercept form or slope-point form.

However, standard form is useful because any linear equation can be written in standard form. However, not every linear equation can be written in slope-intercept form or slope-point form.

So for example, the equation x equals 2 is a line that is perfectly vertical. And so remember that lines that are vertical have a slope that is undefined. So because you have some lines that have slopes that are undefined, these lines cannot be written in slope-intercept form or slope-point form because we don't have a value for m. Our slope is undefined so we don't have a value we can substitute into the equation for m.

So let's go over our key points from today. As usual, make sure you get them in your notes if you don't have them already so you can refer to them later.

The slope and y-intercept of a line can be determined easily by looking at its equation in slope-intercept form. The slope endpoint on a line can be determined easily by looking at its equation in point-slope form. And the equation for any line can be written in standard form, but lines with a slope that is undefined, which are vertical lines, cannot be written in slope-intercept or point-slope form.

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