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Find the minimum value of a + b in the system of equations.
 

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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