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

Step 2: Substitute into the inequality

Substituting into \(x + y < z + 5\):
\((z - 9) + (z - 1) < z + 5\) 
\(2z - 10 < z + 5\) 
Subtract \(z\) from both sides: 
\(z - 10 < 5\) ⇒ \(z < 15\)

Step 3: Find maximum integer value of \(z\)

Since \(z < 15\), the maximum possible integer value of \(z\) is \(14\)

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

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

✅ Final Answer: 23

Answer 2

Given equations:

• Equation 1: \(x + 9 = z\)
• Equation 2: \(y + 1 = z\)
• Equation 3: \(x + y < z + 5\)

Step 1: Express x and y in terms of z

From Equation 1: \(x = z - 9\)

From Equation 2: \(y = z - 1\)

Step 2: Substitute in Equation 3

\((z - 9) + (z - 1) < z + 5\)

\(2z - 10 < z + 5\)

Subtract \(z\) from both sides:

\(z - 10 < 5 \Rightarrow z < 15\)

The maximum possible integer value for \(z\) is \(14\).

Step 3: Maximize \(2x + y\)

Using \(x = z - 9\) and \(y = z - 1\):

\(2x + y = 2(z - 9) + (z - 1) = 2z - 18 + z - 1 = 3z - 19\)

Substitute \(z = 14\):

\(3 \times 14 - 19 = 42 - 19 = \boxed{23}\)

✅ Final Answer: 23