Graphs of linear equations
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Create a graph of the linear equation 5x plus 2y is equal to 20. So the line is essentially the set of all coordinate, all x's and y's, that satisfy this relationship right over here. To make things simpler, what we're going to do is set up a table where we're going to put a bunch of x values in and then figure out the corresponding y value based on this relationship. But to make it a little bit simpler, I'm going to solve for y here. So it becomes easier to solve for y for any given x. So we have 5x plus 2y is equal to 20. If we want to solve for y, let's just get rid of the 5x on the left-hand side. So let's subtract 5x from both sides of this equation. The left-hand side, these guys cancel out, so we get 2y is equal to the right hand side, you have 20 minus 5x. And then you can divide both sides of this equation by 2. So you divide both sides by 2. The left-hand side, we just have a y, and then the right-hand side, we could leave it that way. That actually would be a pretty straightforward way to leave it, or we could call this 20 divided by 2 is 10 minus 5x over 2 or minus 5/2 times x. And so now using this, let's just come up with a bunch of x values and see what the corresponding y values are, and then just plot them. So let me do this in a new color. So let me-- a slightly different shade of yellow. So we have x values, and then let's think about what the corresponding y value is going to be. So I'll start, well, I could start anywhere. I'll start at x is equal to 0, just because that tends to keep things pretty simple. If x is 0, then y is equal to 10 minus 5/2 times 0, which is equal to 5/2 times 0 is just a 0. So it's just 10 minus 0 or 10. So that gives us the coordinate, the point, 0 comma 10. When x is 0, y is 10. So x is 0. So it's going to be right here at the middle of the x-axis. And you go up 10 for the y-coordinate. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. So it's right over here. So that's the point 0 comma 10. Let's do another point. Let's say that x is 2. I'm going to pick multiples of 2 here just so that I get a nice clean answer here. So when x is 2, then y is equal to 10 minus 5/2 times 2, and the 2 in the denominator cancels out with this 2 in the numerator. So it simplifies to 10 minus 5, or just 5. So that tells us the point x equals 2, y is equal to 5, is on the line. So 2x is equal to 1, 2 right over here. And then y is equal to 5. You go up 5. 1, 2, 3, 4, 5, just like that. So that's the point 2, 5. And when you're drawing a line you actually just need two points. If you have a ruler or any kind of straight edge, we could just connect these two points. And if we do it neatly, every point on that line should satisfy this relationship right here. Just so we get practice, I'll do more points. So let me do, let's say when x is equal to 4, then y is equal to 10 minus 5/2 times 4. This is equal to 5/2 times 4. This is equal to 10, right? Because the 2, divide the denominator by 2 you get 1, divide the numerator by 2 you get 2, or 4 over 2 is the same thing as 2. So it becomes 2 times 5 is 10, 10 minus 10 is 0. So the point 4 comma 0 is on our line. So x is 1, 2, 3, 4, and then y is 0. So we don't move up at all, so we have 4 comma 0. And I could keep going. I could try other points. You could do them if you like, but this is plenty. Just two of these would have been enough to draw the line. So let me just draw it. So I'll do it in white. So the line will look something like this. And I could keep going in both directions. So there you have it. That is the graph of our linear equation. Let me make my line a little bit bolder, just in case you found that first line hard to read. So let me make it a little bit bolder. And I think you get the general idea.