Hi, I'm Drew Moyer, and this is an equation for solving for the coordinates of a triangle. To solve for the coordinates, we're gonna need the equations of each of the three lines that make up the triangle. So suppose I have AB is equal to y equals 2x plus one. BC will be y equals negative x minus two. And CA will be y equals one-half x minus two. And to find the points of this triangle, we're gonna need to set up three separate systems of equations, which are used to find the intersection point where two lines will meet. And I'm gonna need to do that three separate times. So point A I know must meet at the intersection of AB and CA. So my system of equations gonna look like this. Y equals 2x plus one, and under that I'm gonna write y equals one-half x minus two. Now there's a couple different ways that you can solve for a system of equations. I'm going to use the substitution method, which you get by knowing that both y values equal two different things, therefor those two different things must equal each other. So I will set up my equation just like this, 2x plus one equals one-half x minus two. And now I'm simply gonna solve for x. First I'm gonna multiply everything by two to get rid of this fraction, 4x plus two equals x minus four. And then I will bring this x over to this side by subtracting it. Positive 3x plus two equals minus four. And then I will bring this two over here by also subtracting it. So we'll get 3x equals negative six, then divide both sides by three, x equals negative two. And there I have the x coordinate of the intersection point of these two lines, the x coordinate of point A. And to get the y coordinate it's very easy, all I have to do is plug this value in to either one of the two equations that I already use. I'm gonna use this one because it's easier. So I have y equals two times negative two plus one, y equals negative four plus one which is negative three. So I know that point A is at negative two, negative three. So I'll graph that negative two, negative three,right there. So now I've got point A. And point B and C are gonna be solved the exact same way. So point B I know is at the point where AB meets BC. So I can set up y equals 2x plus one and y equals negative x minus two. Set them equal to each other, 2x plus one equals negative x minus two. I'm gonna add x to both sides, 3x plus one equals negative two, subtract one from both sides, 3x equals negative three. Divide both sides by three and I get x equals negative one. And then same thing, I take that value and plug it back into the easier equation, y equals two times negative one plus one, which equals negative two plus one, which is negative one. So I know that this point, B, resides at negative one, negative one. Negative one, negative one right here. And again, one more for good measure, I'm gonna find point C by finding the intersection of BC and CA. BC is y equals negative x minus two, and CA is y equals one-half x minus two. I'll set them equal to each other, negative x minus two equals one-half x minus two, and here I see that I have minus twos on both sides. So if I add two to both sides, they will drop out. So now I have negative x equals one-half x, which doesn't make any sense, so I know that x must equal zero. And I plug that value back in to the easier equation, y equals negative zero minus two, which equals negative two. So I know my point C resides at zero, negative two. Zero, negative two, right here. And then, just a game of connect the dots, and I have my triangle. So again, I'm Drew Moyer, and this is an equation for solving for the coordinates of a triangle.