| Literal Equations |
ax + by = c
ax2 + by2 = c
d <= x + y
xy <= c
Find the minimum value of a + b, assuming all variables are positive. |
Problem Moderated by: Sasha |
| Problem Solution |
(ax + by)(x + y - 1) = c(x + y - 1) = (ax2 + by2) + xy(a + b) - (ax + by) = c + xy(a + b) - c = xy(a + b)
x + y - 1 = xy/c(a + b)
x + y = 1 + xy/c(a + b)
d <= 1 + xy/c(a + b)
cd - c <= xy(a + b)
So a + b is at least (cd - c)/xy. Maximizing xy will minimize a + b -- so xy = c, and (a + b)min = d - 1
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