How do you find the exact solutions to the system $y^{2} = x^{2} - 7$ $y^2=x^2-7$ and $x^{2} + y^{2} = 25$ $x^2+y^2=25$ ?

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How do you find the exact solutions to the system $y^{2} = x^{2} - 7$ $y^2=x^2-7$ and $x^{2} + y^{2} = 25$ $x^2+y^2=25$ ?

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Answer 1 · 2016-10-07T14:42:05

Let equation 1 be $y^{2} = x^{2} - 7$ $y^2 = x^2 - 7$ and equation $2$ $2$ be $x^{2} + y^{2} = 25$ $x^2 + y^2 = 25$ .

We know the value of $y^{2}$ $y^2$ in both equations, so we can substitute $y^{2}$ $y^2$ in equation $1$ $1$ , which is already isolated, for $y^{2}$ $y^2$ in equation $2$ $2$ .

$x^{2} + (x^{2} - 7) = 25$ $x^2 + (x^2 - 7) = 25$

$2 x^{2} = 32$ $2x^2 = 32$

$x^{2} = 16$ $x^2 = 16$

$x = \pm 4$ $x = +- 4$

$y^{2} = 4^{2} - 7 AND y^{2} = - 4^{2} - 7$ $y^2 = 4^2 - 7" AND "y^2 = -4^2 - 7$

$y^{2} = 9 AND y^{2} = 9$ $y^2 = 9" AND "y^2 = 9$

$y = \pm (3) AND y = \pm (3)$ $y = +-(3)" AND "y = +-(3)$

$:.$ The solution is $(+-4, +-3)$

Hopefully this helps!

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How do you find the exact solutions to the system $y^{2} = x^{2} - 7$ $y^2=x^2-7$ and $x^{2} + y^{2} = 25$ $x^2+y^2=25$ ?

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How do you find the exact solutions to the system $y^{2} = x^{2} - 7$ $y^2=x^2-7$ and $x^{2} + y^{2} = 25$ $x^2+y^2=25$ ?