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 the equations:
\(x + 9 = z = y + 1\)
and the inequality: \(x + y < z + 5\)

Step 1: Express \(x\) and \(y\) in terms of \(z\)

From \(x + 9 = z\), we derive: \(x = z - 9\) 
From \(y + 1 = z\), we derive: \(y = z - 1\)

Step 2: Substitute into the inequality

Substituting the expressions for \(x\) and \(y\) into \(x + y < z + 5\):
\((z - 9) + (z - 1) < z + 5\) 
Simplifying the left side yields: \(2z - 10 < z + 5\) 
Subtracting \(z\) from both sides: \(z - 10 < 5\)
Adding \(10\) to both sides: \(z < 15\)

Step 3: Determine the maximum integer value of \(z\)

Given that \(z < 15\), the largest possible integer value for \(z\) is \(14\)

Step 4: Calculate \(x\), \(y\), and the required expression

Using \(z = 14\):
\(x = z - 9 = 14 - 9 = 5\) 
\(y = z - 1 = 14 - 1 = 13\) 
The required expression is \(2x + y\):
\(2x + y = 2 \times 5 + 13 = 10 + 13 = \boxed{23}\)

✅ Final Answer: 23