x and y are real numbers that satisfy this system of equations:
x2 + y2 =107
2xy+3x+3y =73
Find the value of x+y
x+y)^2 = 107 + 2xy
2xy = 73 - 3(x+y)
(x+y)^2 = 107 + 73 - 3(x+y)
(x+y)^2 + 3(x+y) - 180 = 0
(x+y) = (-3 ± sqrt(729))/2
(x+y) = (-3 ± 27)/2
(x+y) = {-15, 12}
The graph is of a circle and a hyperbola-like curve one limb of which intersects the circle in the first quadrant at (10.183, 1.817) and (1.817, 10.183) which indeed sum to 12. In the 4th quadrant, the 2 curves do not intersect, but at their closest point of approach, the point midway between them is (-7.5, -7.5) which sum to -15.
If -15 is the sum: complex solution only, but x+y checks out.
x^2 + y^2 = 107
x + y = -15
x^2 + (-15-x)^2 = 107
2x^2 + 30x + 225 = 107
2x^2 + 30x + 118 = 0
x = (-30 ± sqrt(-44))/4
(x,y) = (-7.5 + ((√11)/2)i , -7.5 - ((√11)/2)i)
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Posted by Larry
on 2024-06-16 21:42:16 |
Solution
