[MUSIC PLAYING] Let's look at our objectives for today. We'll start by reviewing how to graph an equation. We'll then look at the standard form of an equation. And finally, we'll look at two different methods for graphing equations in standard form.
Let's start by reviewing graphs of equations. It's often useful to graph equations in order to visually represent the relationship between variables. In order to graph a line, you only need to find at least two points on the line.
There are many ways to graph a line. One strategy might be to pick two values for x and find the corresponding y values to plot on a graph. Another strategy is to write an equation in a certain form to easily identify important information about the line.
Let's look at the standard form of an equation. Equations in standard form look like this. A times x plus B times y equals C. The benefit of having an equation in standard form is that it makes finding the x- and y-intercepts relatively easy. To find the y-intercept, we would substitute 0 for x, which makes the whole Ax term 0, which leaves By equals C, which can be easily solved for y.
Similarly, to find the x-intercept, we substitute 0 for y-- making the By term 0 and leaving Ax equals C, which is easily solved for x. The standard form of a line is important when studying systems of linear equations, which will be done later in this course.
Let's do an example graphing an equation in standard form. We have the equation 3x minus 4y equals 12. In standard form, we can easily identify the x- and y-intercepts, which gives us the minimum two points we need to graph a line. So let's start by finding the x-intercept. The x-intercept is at a point x, 0 where y is 0.
So we can substitute 0 for y into our equation, which gives us 3x minus 4 times 0 equals 12. 4 times 0 is 0, so we have 3x equals 12. We then divide both sides by 3, which gives us x equals 4. So the x-intercept is at the point 4, 0.
Now, let's find the y-intercept. The y-intercept is at a point 0, y where x is 0. So we can substitute 0 for x into our equation, which gives us 3 times 0 minus 4y equals 12.
3 times 0 is 0, so we have negative 4y equals 12. We then divide both sides by negative 4, which gives us y equals negative 3. So the y-intercept is at the point 0, negative 3.
We can now plot both of our intercepts on our graph. The x-intercept is 4, 0, and the y-intercept is 0, negative 3. So we have our two points. Finally, we connect our points to create a line.
Let's do another example to demonstrate a second method for graphing an equation in standard form. We have the equation, negative 2x plus 5y equals 10. Instead of graphing the x- and y-intercepts, we can also rewrite this equation in slope-intercept form and graph the equation using the y-intercept and the slope of the line. Slope-intercept form may be preferred for graphing since y is on one side of the equation and everything else is on the other side.
So to write our equation in slope-intercept form, we need to isolate the y variable. We start by adding 2x on both sides. On the right side, 10 and 2x and not like terms, so we cannot actually combine them. Instead, we have 5y equals 2x plus 10.
Notice that we place the x term in front of the constant term because that's what we want it to look like in slope-intercept form. We then divide both sides by 5. On the right side, we divide both terms by 5.
So we have 2x plus 10 over 5. This simplifies to 2/5x and 2 because 10 divided by 5 is 2. So our final equation is y equals 2/5x plus 2.
To graph, we start at the y-intercept, which is at 2. We then use our slope, 2 over 5, to find a second point. So from the y-intercept, we move up 2 and over 5 to place our second point. Finally, we connect our points to create a line.
Let's go over our key points from today. Make sure you get these in your notes so you can refer to them later. It's often useful to graph equations in order to visually represent the relationship between variables. In order to graph a line, you need to find at least two points on the line.
Equations in standard form look like this, Ax plus By equals C. The benefit of having an equation in standard form is that it makes finding the x- and y-intercepts relatively easy by substituting 0 for x to solve for the y-intercept and 0 for y to solve for the x-intercept. And finally, we could also rewrite an equation in standard form into slope-intercept form to graph.
So I hope that these key points and examples helped you understand a little bit more about graphing a line using standard form. Keep using your notes, and keep on practicing. And soon, you'll be a pro. Thanks for watching.
