First, we simplify 6x+2y=36 into 3x+y=18 by dividing by 2. This means that y=-3x+18.
The sum [tex]x^2+y^2[/tex] can be written as: [tex]x^2+y^2=(x+y)^2-2xy[/tex],
from the binomial expansion formula: [tex](x+y)^2=x^2+2xy+y^2[/tex].
Thus, substituting y=-3x+18 and simplifying we have
[tex]x^2+y^2=(x+(-3x+18))^2-2x(-3x+18)[/tex][tex]=(-2x+18)^2+6x^2-36x=4x^2-72x+18^2+6x^2-36x[/tex][tex]=10x^2-108x+18^2[/tex].
This is a parabola which opens upwards (the coefficient of x^2 is positive), so its minimum is at the vertex. To find x, we apply the formula -b/2a. Substituting b=-108, a=10, we find that x is 108/20=5.4.
At x=5.4, the expression [tex]10x^2-108x+18^2[/tex], which is equivalent to [tex]x^2+y^2[/tex], takes it smallest value.
Substituting, we would find [tex]10x^2-108x+18^2=10(5.4)^2-108(5.4)+18^2=291.6-583.2+324[/tex]=32.4 This is the smallest value of the expression.
For x=5.4, y=-3x+18=-3(5.4)+18=1.8.
Answer: (5.4, 1.8)
