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Forms of Linear Equations

Forms of Linear Equations

Author: Anthony Varela
Description:

Determine the slope and y-intercept of an equation in slope-intercept form.

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Hi, this is Anthony Varela. And today, I'm going to introduce a couple of different forms of linear equations. So we're going to talk about slope-intercept form, point-slope form, and standard form.

So first, I'd like to talk about slope-intercept form. Here is the equation of a line written in slope-intercept form, y equals mx plus b. So here is a more concrete example. y equals 4x minus 8 is the equation of a specific line written in slope-intercept form. And we call this slope-intercept form because we can easily drop two valuable pieces of information-- one, the slope by the variable m, and b, the y-intercept.

So here, 4x minus 8-- I know that this line has a slope of 4. And I know that the y-intercept is negative 8. So how do I know this y-intercept is negative 8? Well, thinking about the y-intercept occurring at x equals 0, when x equals 0, mx is also 0 because anything multiplied by 0 is 0.

So I can just write then y equals b. So I know that this then is the y-intercept and not the x-intercept. So this is slope-intercept form, y equals mx plus b. m is the slope. And b is the y-intercept.

Now, another form of linear equations is point-slope form. So here is point-slope form. It's y minus y1 equals m times x minus x1. And we call this point-slope form because I can easily draw two pieces of information-- once again, the slope of a line by the variable m, and then I also have this x1 and y1 in point-slope form. Well, x1 and y1 is a point that lies on the line.

So here, I am looking at the general equation in point-slope form-- or a specific one, sorry, not general, y minus 3 equals 4 times x plus 2. Now, thinking about then drawing out a specific point on the line and then also the slope from an equation in this form, well, I know that m represents the slope. So I can clearly see that the slope of this line is 4.

Now, what is a point that lies on this line? Well, I'm going to be using x1 and y1. But be careful here. This is a tricky part. The point on this line is negative 2, 3. Now, I could see here that my y coordinate is going to be 3 because I have y minus y1, y minus 3. So clearly, y1 is 3.

But here I have x minus x1. And I have x plus 2. Now, these signs are opposite. So I need to then flip the sign of this positive 2 and make it negative 2. So the point on this line then is negative 2 comma 3.

So our point-slope form is y minus y1 equals m times x minus x1. We can easily identify the slope by the variable m. And we can also easily identify a point on the line using x1 and y1.

The last form that I'd like to talk about is standard form. And the standard form for a line is ax plus by equals c. And so here's an example of a equation of a line written in standard form-- 3x minus 5y equals 2. So a, b, and c are just real numbers. They're coefficients with the case of x and y or a constant term here with c.

Now, the cool thing about standard form is you could relatively easily calculate x and y-intercepts. Because with x and y-intercepts, either x or y is going to equal 0. So that would then sort of make one of these terms disappear if one of them is equal to 0. So you can solve a pretty simple equation.

Another cool thing about equations in standard form is that every line can be written in standard form. Not every line can be written under other forms. And you're thinking why might that be? Well, let's consider then this graph here of a vertical line.

So when I identify two points on this line, and I want to calculate the rise and the run. Well, I can clearly see that there's some type of rise from one point to another, but the run is 0. There is no horizontal displacement. And you can't divide by 0, so the slope is undefined.

Now, I can write this then as just x equals 5. But really, what this is 1x minus 0y equals 5. Whereas our slope-intercept form and our point-slope form relies on a defined slope to make the equation, standard form doesn't, which I think is neat. So the standard form is ax plus by equals c. And any line can be written in standard form. Can't say that about the other two forms.

So let's review the forms of linear equations. We have slope-intercept form, y equals mx plus b. m is the slope. And b is the y-intercept. You're really going to want to use equations in this form if you want to know the slope, and you want to know the y-intercept. You can just look at the equation and figure it out.

With point-slope form, this is y minus y1 equals m times x minus x1. m is the slope. And x1, y1 is a point that lies on the line. So you're going to prefer this form when you want to easily identify the slope or easily identify a point on the line.

And then we have standard form ax plus by equals c. And the cool thing about standard form is that any line can be written in standard form. Doesn't rely on a defined slope. And you could also use this to relatively easily calculate x and y-intercepts. So thanks for watching this tutorial on the forms of linear equations. Hope to see you next time.

Formulas to Know
Point-Slope Form of a Line

y minus y subscript 1 equals m left parenthesis x minus x subscript 1 right parenthesis

Slope-Intercept Form of a Line

y equals m x plus b

Standard Form of a Line

A x plus B y equals C