Solve the nonlinear system of equation for real solution

5 x

^{2}+ 3 y^{2 = 120}-x

^{2 }+ y^{2 = 16}a) The solution is ???????

b)the is no solution

Solve the nonlinear system of equation for real solution

5 x^{2} + 3 y^{2 = 120}

-x^{2 }+ y^{2 = 16}

a) The solution is ???????

b)the is no solution

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Ansonia, CT

There are four; the first equation is an ellipse, and the second is a hyperbola that intercepts the ellipse at (±3, ±5). Try substitution as follows:

y^{2} = 16 + x^{2}, so 5x^{2} + 3(16 + x^{2}) = 120 and solve as

5x^{2} + 48 + 3x^{2} = 120

8x^{2} = 120 - 48

x^{2} = 72/8 = 9 so x = ±3

either of these values plugged into the second equation yields y^{2} = 16 + 9 = 25 so y = ±5

the full set would be (-3, -5), (-3, 5), (3, -5), (3, 5)

Good Luck!

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