Writing Linear Equations Given Two Points In Standard Form Point Slope Form

in this video we're going to talk about how to write linear equations particularly if you're given the slope and point if you're given two points if you're given data table and we're going to talk about how to put it in its three different forms slope intercept form point slope form and in standard form so the first kind of question that you might see is write the linear equation given the slope let's say it's two and the intercept 03 if you're given slope and intercept the best thing to do is to put it in slope intercept form using the equation is equal to MX plus the intercept is the value so all you need to do is replace and is 2 is -3 so the equation in slope intercept form is = 2x - 3 let's try another example so let's say if you're given the slope 1 over3 and the intercept negative actually positive 5 write the equation in slope intercept form so all you need to do is replace and that's it for this type of problem so the answer is is equal to 1 over 3x + 5 now let's talk about how to write linear equation given two points let's say the two points are 1A 3 and 3 comma 7 now the first thing that you need to do is find slope between these two points the equation that you need is this the slope is equal to the change in the values Y2 - y1 divided by the change in the values X2 - X1 so X1 is the coordinate of the first point and 3 is y1 the second point is going to have X2 that's the coordinate for the second Point Y2 is the y-coordinate for the second point so now we just got to plug it into the formula Y2 is 7 y1 is 3 X2 is 3 X1 is 1 7 - 3 is 4 3 - 1 is 2 4 / 2 is 2 so now we have slope of two once you have the slope then you can write the equation in standard form or slope intercept form or any form you need to in this example we're going to write the equation first in point slope form here's the formula if you want to write it in point slope form y1 is three is 2 X1 is 1 you can use this point or you can use the other point it doesn't matter you're going to get the same answer so this is the answer in point slope form now if you want the answer in slope inter acept form you need to get by itself on one side of the equation so first let's distribute the two 2 - 1 so it's going to be 2x - 2 next let's add three to to both sides so is = 2x -2 + 3 is 1 so this is the answer in slope intercept form now let's convert it to standard form in standard form the and the variables have to be on the same side so we need to move the 2x to the left side if we subtract both sides by 2x we're going to have -2x + 1 is equal to 1 now like to get rid of this negative sign so I'm going to multiply both sides by negative 1 so I'm going to change the sign of everything so we're going to have POS 2x - 1 or simply minus is equal to1 so this is the linear equation in standard form so now you're familiar with all three forms the first one that we talked about is the point slope form which you can use this equation next we have the slope intercept form which is = mx plus and finally standard form ax + yal so make sure you're aware of these three different forms now let's work on another example let's say is -2 and is 1 and is three and let's say is 16 go ahead and write the linear equation in slope intercept form you don't have to put it in other forms just get it into slope intercept form now I'm going to use method that's different from the last example I'm going to use the slope intercept form directly but first we need to calculate the slope so this is X1 y1 X2 Y2 so the slope is 2 - y1 / X2 - X1 2 is 16 y1 is 1 X2 is 3 X1 is -2 16 - 1 is 15 3 - -2 is the same as 3 + 2 which is 5 and 15 / 5 is three so the slope is three now let's use the slope intercept form formula equal MX plus now I'm going to choose the second Point 316 so what we need to do if you're going to use this formula directly you need to find the value of we have the value of is 16 We Know is 3 and is also 3 3 * 3 is 9 and to find it we got to subtract both sides by 9 so 16 - 9 is 7 once you have and you can now write the equation in slope intercept form all you need to do is replace and is 7 so the answer in this example is is = 3x + 7 now you also need to be able to write linear equations if you're given graph given this particular graph write linear equation in slope intercept form if you could find slope in the intercept then you can easily write it notice that the intercept step where the graph touches the AIS is -3 so therefore we know that is -3 now to find slope you can use rise over run the slope is the rise which is the change in values Y2 - y1 divided by the run which is the change in values X2 - X1 so to go from the first point to the second point you need to rise three units and and run four units so it's rise over run it's 3 over 4 now that we have the slope and the intercept we can write the equation of the line so replace with three over 4 and replace with -3 so this is the answer in slope intercept form is equal to 3 over 4 - 3 let's look at another example this time go ahead and write the equation in standard form let's say it passes through these two points so notice that the intercept is three so is equal to three in this example and now let's find slope the rise it's going down so the rise isg -3 the run is five units so the slope is rise over run -3 / 5 the equation in slope intercept form is going to be -3 over 5x + 3 so that's the answer in slope intercept form but we want to get it in standard form so we need to get the and the variables on one side of the equation to do that let's get rid of the fraction let's multiply everything by 5 so we're going to have 5 is equal to 5 * -3 over 5 the fives will cancel leaving behind -3x and then we need to multiply 3 by 5 which is positive 15 now all we need to do at this point is ADD 3x to both sides so 3x + 5 is equal to POS 15 and so this is the answer this is the linear equation in standard form and this is the answer in slope intercept form if you need it to be in that form now sometimes you might be given data table and let's say the data on this table represents linear equation how can you use use this information to write linear equation first in point slope form and then in slope intercept form now all you need to do is pick two out of the many points that are in this table and you can just write the linear equation based on previous example that we covered earlier in the video so let's focus on these two points so my first ordered pair is going to be 5 comma 3 the second is 7 comma 7 so this is X1 y1 X2 Y2 so therefore the slope is going to be Y2 - y1 ided by X2 - X1 so Y2 is 7 y1 is 3 X2 is 7 X1 is 5 7 - 3 is 4 7 - 5 is 2 4 / 2 is 2 so the slope is positive2 now that we have the slope we can write the equation in point slope form so let's use this formula - y1 is equal to * - X1 y1 is 3 is 2 X1 is 5 so I'm using this point but you can use any one of the four points you can use this point you can use that one it's going to work you'll get the same answer so this is the equation in point slope form but now let's get the answer in slope intercept form so let's distribute the two it's going to be 2X and then 2 * -5 that's -10 now let's add three -10 + 3 is -7 so this is the answer in slope intercept form so is equal to 2x - 7 and that's it