Question

The number of pairs of integers \((x , y)\) satisfying \(x|\geq y\geq-20\) and \(2x+5y=99\) is

Answer 1

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

From the given equation: \[ 2x + 5y = 99 \Rightarrow x = \frac{99 - 5y}{2} \] For \(x\) to be an integer, \(99 - 5y\) must be even. Since 99 is odd and 5y is odd when y is odd, we conclude: \[ y \text{ must be odd} \]

Step 2: Find valid integer values for \(y\)

Try \(y = -19\), then: \[ x = \frac{99 - 5(-19)}{2} = \frac{99 + 95}{2} = \frac{194}{2} = 97 \] Try \(y = 13\), then: \[ x = \frac{99 - 65}{2} = \frac{34}{2} = 17 \] So, valid \(y\) values range from \(-19\) to \(13\) in steps of 2 (odd numbers).

Step 3: Count the number of valid integer pairs

This is an arithmetic progression with: \[ a = -19,\quad d = 2,\quad t_n = 13 \] Use the formula for nth term: \[ t_n = a + (n - 1)d \] \[ 13 = -19 + (n - 1) \cdot 2 \Rightarrow 32 = 2(n - 1) \Rightarrow n = 17 \] So, there are 17 valid values of \(y\), and each gives an integer \(x\), satisfying \(x \geq y\).

✅ Final Answer: 17 integer pairs

Answer 2

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

Rearranging the equation: \[ x = \frac{99 - 5y}{2} \] For \(x\) to be an integer, the numerator must be even. Since 99 is odd, \(5y\) must also be odd ⇒ \(y\) must be odd.

Step 2: Explore values of \(y\)

Let’s try some values of odd \(y\):

• If \(y = -19\), \(x = \frac{99 + 95}{2} = 97\)
• If \(y = 13\), \(x = \frac{99 - 65}{2} = 17\)
• If \(y = 15\), \(x = \frac{99 - 75}{2} = 12\)

We observe that each increment of \(y\) by 2 decreases \(x\) by 5.

 

Step 3: Determine range of \(y\)

Let’s find all odd values of \(y\) such that \(x = \frac{99 - 5y}{2}\) is an integer and \(x \geq y\). 

We are told: \[ x \geq y \Rightarrow \frac{99 - 5y}{2} \geq y \Rightarrow 99 - 5y \geq 2y \Rightarrow 99 \geq 7y \Rightarrow y \leq 14.14 \] So the largest valid odd integer \(y\) is 13. 
Also: \[ \text{Smallest odd integer that gives integer } x = -19 \Rightarrow \text{Start from } y = -19 \text{ up to } y = 13 \]

Step 4: Count the odd values between -19 and 13

This is an arithmetic sequence: \[ a = -19, \quad d = 2, \quad l = 13 \] Number of terms: \[ n = \frac{(13 - (-19))}{2} + 1 = \frac{32}{2} + 1 = 16 + 1 = 17 \]

✅ Final Answer: 17 integer pairs