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Video transcript

we have the system of equations y is equal to 4x minus 17.5 and y plus 2x is equal to 6 point 5 and we have to solve for x and y so we're looking for X's and Y's that satisfy both of these equations now the easiest way to think about it is we've already solved for y in this top equation let me write it again I'll write it in pink we have y is equal to 4x minus 17.5 so this first equation is telling us literally by this constraint Y should be 4 times X minus 17 and 1/2 now the second equation says whatever Y is we add 2 times X and that should be 6 point 5 well the y here also has to meet this constraint up here it also has to meet the constraint that it has to be 4 times X minus 17.5 so what we can do is is we can substitute this value for Y into this equation let me be clear what I'm doing the second equation here is y plus 2x is equal to 6 point 5 we know we know that Y has to be equal to this thing right here Y has to be equal to 4x minus 17.5 so let's take 4x minus 17.5 and substitute Y with that so let's put that right there so if we were to do that if we were to replace this Y with 4x minus 17.5 because that's what the first equation is telling us then we get 4x minus 17.5 plus plus 2x is equal to 6.5 and now we have a single linear equation with one unknown let's solve for X so first we have our X terms we have a 4x and we have a 2x we can group them or add them together 4x plus 2x is 6x and then we have 6x minus 17.5 - 17.5 is equal to is equal to 6.5 then we can get the 17.5 out of the way by adding it to both sides of the equation so this is negative 17.5 so let's add positive 17.5 to both sides of this equation and we are left with the left-hand side is just going to be 6x because these guys cancel out 6x is going to be equal to and 6.5 z6 plus 17 is 23 and then 0.5 plus 0.5 is 1 so this is going to be 24 and then we can divide both sides of this equation by 6 both sides by 6 and you are left with X is equal to X is equal to 24 over 6 which is the same thing as 4 so we figured out the x value that for the X&Y pair that satisfy both of these equations now we need to figure out the Y value and we can do that by taking this X and putting it back into one of these equations we can do it into either one we should get the same y value so let's just do this top one up here so if we assume X is equal to 4 this top equation tells us that Y is equal to 4 4 times X which in this case is 4 minus 17.5 - 17.5 well this is equal to 16 minus seventeen point five which is equal to negative one point five so Y is equal to negative point one point five so the solution to this system is X is equal to 4 y is equal to negative one point five and you can even verify that these two they definitely work for the top one if you put 4 times 4 minus 17.5 you get negative one point five but they also work for the second one and let's do that in the second one if you take negative 1.5 plus 2 times X plus 2 times 2 times 4 what is that equal that's negative 1.5 plus 8 well negative 1 point 5 plus 8 is 6 point 5 so this x and y satisfy both of these equations