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Finding the Equation of a Graphed Line - Problem 2
To write the equation of a line, first find the slope. Remember that slope is the change in y over the change in x. To find slope, find two points on the line. Then use either the slope formula (y2-y1)/(x2-x1) or use the slope triangle. To use a slope triangle, connect the two points by drawing a triangle. The length of the vertical leg is the numerator of the slope (change in y, the rise) and the length of the horizontal leg is the denominator of the slope (change in x, the run). If the line does not have a whole number y-intercept, use the point-slope form of a line y-y1=m(x-x1) to find the equation of the line. Plug in the slope and one coordinate point from the line and solve for y.
Here I'm given a graph that has two points labelled on it and I'm asked to write the equation for that graphed line. So the first thing I'm going to want to figure out is what the slope is of the line.
Slope is the change in y on top of change in x. So one strategy you could do is to use this formula where slope is the difference of your y terms on top of the difference of your x terms.
A shortcut way that I like a little bit better is to draw a slope triangle however if you do this in your textbook make sure you use pencil as you can erase it and note get in trouble.
Okay slope triangle looks like this if you have your two points you connect them using a triangle, label the sides and I can see my slope is going to be negative one third.
I didn't have to use this formula in those points I could have but I already know my slope number is going to be -1/3. Okay so now to write the equation I have a choice I could use y equals mx plus b form or I could use this y minus y1 equals slope times x minus x1 form. Either one of those are valid forms of an equation.
So for me I'm going to try to use this one because it looks a little bit easier let's check out the graph and see if we can find the y intercept. If you look the graph that this problem gave me doesn't show where it crosses the y axis so want to take a ruler and try to extend that line so I can see it.
This might work for you let's see. If I extend this line oh-oh! I can see that it crosses at some kind of fraction. This line goes on for ever and ever so it does cross the y axis just in this picture I can tell it's going to be some kind of fractional answer. It looks like less than a half I don't know if it is like a quarter maybe a third maybe something like two fifths.
I don't know so this equation form might not be the best for this problem because I'd have to approximate that y intercept. I'm not going to do that. I want to get the problem right okay. So I'm going to try using this other formula which isn't so bad it just involves quite a bit of algebra.
So for this point slope form of the equation I can choose either point I want to and use my slope value of negative -1/3 to find the equation. I'm going to choose to use this point (-5,5) I just chose that. You would get the same answer eventually if you use (-2,4) it shouldn't matter.
Okay so y minus my y value is equal to my slope number times x minus my x value. Be careful I have minus, minus 5 or minus -5 which is going to become a +5 right there.
Let me rewrite this so I have more space and then we'll go through and solve for y. Although I don't actually have to solve for y what I have here is already an equation for a one that's the point slope equation for that line.
I just because I personally like slope intercept I want to go through and solve this a little bit further but depending on your teacher or your text book you might be done there.
Let me show you how I will continue. Y take away 5 is equal to one third, x plus 5, okay let's go through and distribute that -/3. -1/3x minus 5/3, look out for the fractions especially when I'm adding five to both sides. If I'm adding 5 to -5/3 I want to find the common denominator. The common denominator between 5 and -5/3 is going to be 3 so I need to turn 5 into 15/3.
So now what I'm doing is -5/3 plus 15/3 which is going to give me positive 10/3. That's ugly. Good thing I did the algebra to find out what that intercept was because using the line there I wouldn't have been able to find out precisely that that arrow crossed one third of the line that was the value y equals 3.
So here we go this is the same equation written in two different forms so be careful to read the directions and see what forms they are asking for if you're lucky it will ask for the equation in point slope form and you'll be done there, if it asks for Slope-Intercept form that's where you have to continue solving to get to here.