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First, we simplify 6x+2y=36 into 3x+y=18 by dividing by 2. This means that y=-3x+18.
The sum
can be written as: 
,
<span>
from the binomial expansion formula: </span>
.
<span>
Thus, substituting </span>y=-3x+18 and simplifying we have<span>
</span>
.
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
, which is equivalent to , takes it smallest value.
Substituting, we would find
=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)
The sum
<span>
from the binomial expansion formula: </span>
<span>
Thus, substituting </span>y=-3x+18 and simplifying we have<span>
</span>
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
Substituting, we would find
For x=5.4, y=-3x+18=-3(5.4)+18=1.8.
Answer: (5.4, 1.8)