Question

If x and y are non-negative integers such that \(x + 9 = z, y + 1 = z\) and \(x + y < z + 5\), then the maximum possible value of \(2x + y\) equals

Answer 1

Given \(x+9=z=y+1\) and \(x+y<z+5\)

\(⇒ (z-9)+(z-1)<z+5\)

\(⇒ z<15\)

Hence the maximum value of  \(z=14\), max of \(x=5\) and max of \(y=13\)

Required answer, \(2x+y=2×5+13=23\)

Answer 2

Given:
Equation 1. \(x+9=z\)
Equation 2. \(y+1=z\)
Equation 3. \(x+y\lt z+5\)
Now From equation 1 and 2:
\(x=z-9\)
\(y=z-1\)
Now put these expressions for x and y in equation 3
\((z-9)+(z-1)\lt z+5\)
\(2z-10\lt z+5\)
\(z\lt15\)
Now, the maximum possible value can't be 15 or greater than 15. So, less than 15 is 14.
\(z=14\)
We need to maximize
\(2x+y\)
\(2(z-9)+(z-1) \;=\;3z-19\)
Now put the value of z
\(3(14)-19 =23\)

So, the answer is 23.