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CCSS Math: HSA.REI.C.7

What are the solutions to
the system of equations y is equal to 1/2 x and 2x squared
minus y squared is equal to 7. And they say solutions,
because these two, if we view them as 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
and 2x squared minus y squared is equal to 7. And the best way
to approach these is to 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 minus y squared. But now we're 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 minus. So 1/2 squared is 1/4,
and then x squared. So we could say x squared
over-- let me write that as 1/4. So let's say it's 1/4 x
squared is equal to 7. So I have 2x squareds, 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. Multiply both sides by 4/7. And we get x squared
is equal to 4. And so x could be
positive or negative 2. 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 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, then y is going to be 1/2 times
negative 2, which is going to be negative 1. And 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 whole thing's
going to be 8. Minus negative 1
squared still equals 7. So the solutions to the
system of equation-- one is the coordinate 2
comma 1. x is 2. y is 1. The other is the
coordinate negative 2, negative 1. x is negative
2, and y is negative 1.