How do you find the exact solutions to the system #5x^2+y^2=30# and #9x^2-y^2=-16#?

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Answer 1 · 2016-11-13T15:20:54

The solution set is #{-1, 5}; {-1, -5}; {1, 5}; {1, -5}#

#y^2 = 30 - 5x^2 -> 9x^2 - (30 - 5x^2) = -16#

#9x^2 - 30 + 5x^2 + 16 = 0#

#14x^2 - 14 = 0#

#14(x^2 - 1) = 0#

#x^2 - 1 = 0#

#(x + 1)(x- 1) = 0#

#x = -1 and 1#

Substitute back into one of the original equations.

#5x^2 + y^2 = 30#

#5(-1)^2 + y^2 = 30 and 5(1)^2 + y^2 = 30#

#5 + y^2 = 30 and 5 + y^2 = 30#

#y^2 = 25 and y^2 = 25#

#y = +-5 and y = +-5#

So, the solution sets are as follows:

#{-1, 5}; {-1, -5}; {1, 5}; {1, -5}#

Hopefully this helps!