How To Find The Equation of a Line From a Graph Algebra

in this video we're going to talk about how to find the equation of line from graph so you need to be familiar with this formula is equal to MX plus it's linear equation in slope intercept form the slope is the number in front of represented by the letter the y-intercept is that's where the graph touches the y-axis so let's go ahead and begin let's start with graph number one the red graph what is the y-intercept where does the line touch the y-axis well this is the x-axis this is the y-axis the red line touches it at this point which is at 2. so we could say is equal to two let's find the y-intercepts for the other graph so for the blue graph the y-intercept is for the green one it's negative two so that's how you could find when the y-intercepts now let's calculate the slope the slope is rise over run so we need at least two points in order to calculate the slope so let's focus on these two points here in order to go from the first point to the second point we need to rise by two units and we need to run to the right by three units this is at negative three so we have rise of two run of three so the slope rise over run it's two over three and it's positive because it's going up now once we know the values of and we now have enough information to write the equation of the line all we got to do is replace and in that formula so it's going to be is equal to 2 over 3x Plus 2. so that's the answer for the first example now let's move on to the second example graph number two the blue graph we have the y-intercept but what is the slope for that graph feel free to pause the video if you want to try this problem you could also work on the third graph as well so let's begin by identifying two points that we want to focus on let's use those two points so what is the rise and what is the Run in order to go from the first point to the second point we need to go down by three units so it's negative three because we're going down so the run mean the rise rather the rise is negative three if we're going up it would be positive since we're going down it's negative now what is the Run as we travel horizontally to get to the second point we need to go four units to the right so the run is four so we have slope of negative three over four so we have y-intercept of three and we have slope so now we can just replace and we're going to get the answer so for the second example it's going to be is equal to negative three over four times plus the intercept so that's how you can get the equation of the line from the graph now let's look at our last example so we already have the intercept let's get the slope let's focus on these two points so we need to rise 5 units and we need to run only two units to the right I'm going to draw that better so here's the rise up five over two so we have slope of 5 over 2. so now we can write the linear equation is equal to MX plus 5 over 2x plus negative 2 or simply minus 2. so that's the equation of the line for the second example now sometimes if you're if you have multiple choice exam the answer may not always be written in slope intercept form sometimes the answer may be written in standard form other times even point-slope form so you need to know how to change it from one form to another so let's say we want to write this in standard form that is an ax plus equals format how can we do that the first thing would do is get rid of the fractions would multiply everything by 3. so we'll have 3y is equal to 3 times 2 3 plus 2 times 3. the threes will cancel so we'll have 3y is equal to 2X and 2 times 3 is 6. now we could take the 2X and move it to the other side it will change from positive 2x but on the left side it will be negative 2X and so we have this negative 2x plus 3y equals 6. if you want to you can multiply everything by negative one and you'll get positive 2x minus 3y is negative 6. so it can be written in any one of these two forms but that's how you can convert it from slope intercept form to standard form but now what about converting it from slope intercept form to point-slope form how can we do that let's use this one as an example point-slope form looks like this it's minus y1 is equal to times minus X1 now there's many ways in which you can write this it's not limited to one way but here's something you can do we already know the slope it's negative three over four let's pick point value for the value for I'm going to choose is going to be either 4 or multiple 4. because of this denominator so I'm going to choose my value as 4. if plug in 4 what will get for 4 divided by 4 is 1. I'll get negative 3 plus 3 is 0. don't want zero want something other than zero let's try plugging in 8. 8 divided by 4 is 2. and 2 times negative 3 is negative 6. negative six plus three is negative 3. so have the point eight comma negative three when we plug in value of 8 we get negative three for wanted to avoid any fractional points or any fractional values in or so now how do we use this so we have we have X1 and y1 all we got to do is replace these three variables so it's going to be minus y1 which is negative 3 equals is negative 3 over 4. times minus X1 which is 8. so the two negatives will become positive so we have Plus 3. and it's equal to this so now we have the same linear equation but in different format we've converted it from mean we convert it from slope intercept form to point-slope form and you need to be able to do that because when you're taking an exam you may have the right answer in slope intercept form but the answer choices may not show this answer it may show the answer in point slope form or the other one is standard form and you need to be able to convert it from one form to another