Literal Equations

ax + by = c

ax² + by² = c

d
xy
Find the minimum value of a + b, assuming all variables are positive.

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Literal Equations 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 =

(a + b)

x + y = 1 +

(a + b)

d
cd - c
So a + b is at least

(cd - c)

. Maximizing xy will minimize a + b -- so xy = c, and (a + b)_min = d - 1

Conics

Pythagorean Triples

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