Solve equation????

Question

Solve the nonlinear system of equation for real solution

 

5 x² + 3 y^{2 = 120}

 

-x² + y^{2  = 16}

 

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

 

b)the is no solution

 

Answer 1

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:

 

y² = 16 + x², so 5x² + 3(16 + x²) = 120 and solve as

 

5x² + 48 + 3x² = 120

8x² = 120 - 48

x² = 72/8 = 9 so x = ±3

 

either of these values plugged into the second equation yields y² = 16 + 9 = 25 so y = ±5

 

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

 

Good Luck!