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Video transcript

what are the solutions to the system of equation system of equations y is equal to one-half x + 2 x squared - y squared is equal to 7 and they say solutions because this could these two if we view them as two line or two curves they could very well intersect with each other twice so let's see what's going on here we have y is equal to 1/2 x + 2 x squared - y squared is equal to 7 and the best way to approach these is just try to substitute one constraint into the other constraint or substitute one equation into the other one it seems easier to substitute 1/2 X for Y into this equation because they've already solved for y here here it's much harder to solve for X or Y so let's do that every place we see a y here let's substitute it with this that Y must also be equal to 1/2 X and then see if we can solve for X so on this equation we have 2x squared - y squared but now we're saying that Y we're now saying that Y must also be equal to 1/2 X and that is going to be equal to 7 now let's see if we can solve for X doing a little bit of algebraic manipulation so we get 2x squared - so 1/2 squared is 1/4 and then x squared so we could say x squared / let me write that as 1 for it so let's say it's 1/4 1/4 x squared is equal to 7 so I have 2 X Squared's and I subtract out a 1/4 x squared so I'm going to have a 1 and 3/4 x squared or you could view this as 8/4 minus 1/4 is 7/4 x squared 7/4 x squared is equal to 7 multiply both sides times the reciprocal of 7/4 so 4/7 times multiply both sides by 4 sevenths and we get we get x squared is equal to 4x squared is equal to 4 and so X could be positive or negative - it's the positive and negative square root of 4 so X is equal to the plus or minus square root of 4 X is equal to positive 2 positive 2 or negative 2 now given that X is positive 2 or negative 2 let's substitute back into either of these equations to figure out what Y is what the corresponding Y is for each of these so if X is 2 y is going to be 1/2 of that it's going to be 1 so we have the point 2 comma 1 all I did is say look X is 2 1/2 times 2 is 1 if X is negative 2 if X is negative 2 then Y is going to be 1/2 times negative 2 which is going to be negative 1 and you can both of these definitely satisfy this first constraint and you can verify that they also satisfy this second constraint right over here 2 times 2 squared is well 2 squared is 4 2 times that is 8 minus 1 squared is 7 2 times negative 2 squared is still this holder is going to be 8 minus negative 1 squared still equals 7 so the solutions to the system of equation 1 is the coordinate 2 comma 1 X is 2 y is 1 the other is a coordinate negative 2 negative 1 X is negative 2 and Y is negative 1