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CCSS.Math: ,

what are the solutions to the system of equations y is equal to X plus 1 and x squared plus y squared is equal to 25 so let's first just visualize what we're trying to do so let me try to roughly graph these two these two equations so my y axis this is my x axis this is my x axis this right over here x squared plus y squared is equal to 25 that's going to be a circle centered at 0 with radius radius 5 you don't have to know that to solve this problem but it helps to visualize it so if this is 5 this is 5 5 5 this right over here is negative 5 this right over here is negative 5 this equation would be represented by this set of points or this is a set of points that satisfy this equation so let me there you go trying to draw it as close to a perfect circle as I can and then y equals x plus 1 y equals x plus 1 is a line of slope 1 with a 1 y intercept so this is 1 2 3 4 y intercept is there it has a slope of 1 so it looks something looks something like this so when we're looking for the solutions we're looking for the points that satisfy both the points that satisfy both are the points that sit on both so it's that point we do it in green it's this point and it's this point right over here so how do we actually figure that out well the easiest way is to well sometimes the easiest way is to substitute one of these constraints into the other constraint and since they've already solved for y here we can substitute Y in the blue equation with X plus 1 with this constraint right over here so instead of saying x squared plus y squared equals 25 we can say x squared plus and instead of writing a y we're adding the constraint that Y must be X plus 1 so x squared plus X plus 1 squared squared must be equal to 25 and now we can attempt to solve for X so we get x squared plus now we Square this will get right in magenta we'll get x squared plus 2x plus one and that must be equal to 25 we have 2x squared now I'm just combining these two terms 2x squared plus 2x plus 1 is equal to 25 now we could just use the quadratic formula to find the the well we have to be careful we have to set this equal to 0 and then use the quadratic formula so let's subtract 25 from both sides and you could get 2x squared plus 2x minus 24 is equal to 0 and actually let's just to simplify this let's divide both sides by 2 and you get x squared plus X minus 12 is equal to 0 and actually we don't even have to use a quadratic formula we can factor this right over here what are two numbers when we take their product we get negative 12 and when we add them we get positive 1 well positive 4 a negative 3 would do the trick so we have X plus 4 times X minus 3 is equal to 0 so X could be equal to well if this is if X plus 4 is 0 then that would make this whole thing true so X could be equal to negative 4 or X could be equal to positive 3 so this right over here is a situation where X is negative 4 this right over here is a situation where X is 3 so we're almost done we just have to find the corresponding Y's and for that we can just resort to the simplest equation right over here y is X plus 1 so in this situation when X is negative 4 y is going to be that plus 1 so Y is going to be negative 3 y is going to be negative 3 this is the point negative 4 comma negative 3 likewise when X is 3 y is going to be equal to 4 so this is the point 3 comma 4 these are the two solutions to this nonlinear system of equations