Let `x, y in N and 7x+12y=220`. The number of solutions :

Options

A. 1
B. 2
C. 3
D. infinitely many

Answer

To solve the equation \( 7x + 12y = 220 \) for natural number solutions, we will follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ 7x + 12y = 220 \]
We can rearrange this to express \( y \) in terms of \( x \):
\[ 12y = 220 - 7x \]
\[ y = \frac{220 - 7x}{12} \]

**Hint:** Rearranging the equation helps us isolate one variable, making it easier to analyze.

### Step 2: Finding Natural Number Solutions
Since \( x \) and \( y \) must be natural numbers (positive integers), we need \( 220 - 7x \) to be a positive integer that is divisible by 12. This means:
\[ 220 - 7x > 0 \implies 220 > 7x \implies x < \frac{220}{7} \approx 31.43 \]
Thus, \( x \) can take values from 1 to 31.

**Hint:** Determine the range of \( x \) based on the condition that \( y \) must be positive.

### Step 3: Checking Values of \( x \)
We will check values of \( x \) starting from 1 up to 31 to see when \( y \) is a natural number.

1. **For \( x = 1 \):**
\[ y = \frac{220 - 7(1)}{12} = \frac{213}{12} \quad \text{(not a natural number)} \]

2. **For \( x = 2 \):**
\[ y = \frac{220 - 7(2)}{12} = \frac{206}{12} \quad \text{(not a natural number)} \]

3. **For \( x = 3 \):**
\[ y = \frac{220 - 7(3)}{12} = \frac{199}{12} \quad \text{(not a natural number)} \]

4. **For \( x = 4 \):**
\[ y = \frac{220 - 7(4)}{12} = \frac{192}{12} = 16 \quad \text{(natural number)} \]

5. **For \( x = 5 \):**
\[ y = \frac{220 - 7(5)}{12} = \frac{185}{12} \quad \text{(not a natural number)} \]

6. **For \( x = 6 \):**
\[ y = \frac{220 - 7(6)}{12} = \frac{178}{12} \quad \text{(not a natural number)} \]

7. **For \( x = 7 \):**
\[ y = \frac{220 - 7(7)}{12} = \frac{171}{12} \quad \text{(not a natural number)} \]

8. **For \( x = 8 \):**
\[ y = \frac{220 - 7(8)}{12} = \frac{164}{12} \quad \text{(not a natural number)} \]

9. **For \( x = 9 \):**
\[ y = \frac{220 - 7(9)}{12} = \frac{157}{12} \quad \text{(not a natural number)} \]

10. **For \( x = 10 \):**
\[ y = \frac{220 - 7(10)}{12} = \frac{150}{12} \quad \text{(not a natural number)} \]

11. **For \( x = 11 \):**
\[ y = \frac{220 - 7(11)}{12} = \frac{143}{12} \quad \text{(not a natural number)} \]

12. **For \( x = 12 \):**
\[ y = \frac{220 - 7(12)}{12} = \frac{136}{12} \quad \text{(not a natural number)} \]

13. **For \( x = 13 \):**
\[ y = \frac{220 - 7(13)}{12} = \frac{129}{12} \quad \text{(not a natural number)} \]

14. **For \( x = 14 \):**
\[ y = \frac{220 - 7(14)}{12} = \frac{122}{12} \quad \text{(not a natural number)} \]

15. **For \( x = 15 \):**
\[ y = \frac{220 - 7(15)}{12} = \frac{115}{12} \quad \text{(not a natural number)} \]

16. **For \( x = 16 \):**
\[ y = \frac{220 - 7(16)}{12} = \frac{108}{12} = 9 \quad \text{(natural number)} \]

17. **For \( x = 17 \):**
\[ y = \frac{220 - 7(17)}{12} = \frac{101}{12} \quad \text{(not a natural number)} \]

18. **For \( x = 18 \):**
\[ y = \frac{220 - 7(18)}{12} = \frac{94}{12} \quad \text{(not a natural number)} \]

19. **For \( x = 19 \):**
\[ y = \frac{220 - 7(19)}{12} = \frac{87}{12} \quad \text{(not a natural number)} \]

20. **For \( x = 20 \):**
\[ y = \frac{220 - 7(20)}{12} = \frac{80}{12} \quad \text{(not a natural number)} \]

21. **For \( x = 21 \):**
\[ y = \frac{220 - 7(21)}{12} = \frac{73}{12} \quad \text{(not a natural number)} \]

22. **For \( x = 22 \):**
\[ y = \frac{220 - 7(22)}{12} = \frac{66}{12} \quad \text{(not a natural number)} \]

23. **For \( x = 23 \):**
\[ y = \frac{220 - 7(23)}{12} = \frac{59}{12} \quad \text{(not a natural number)} \]

24. **For \( x = 24 \):**
\[ y = \frac{220 - 7(24)}{12} = \frac{52}{12} \quad \text{(not a natural number)} \]

25. **For \( x = 25 \):**
\[ y = \frac{220 - 7(25)}{12} = \frac{45}{12} \quad \text{(not a natural number)} \]

26. **For \( x = 26 \):**
\[ y = \frac{220 - 7(26)}{12} = \frac{38}{12} \quad \text{(not a natural number)} \]

27. **For \( x = 27 \):**
\[ y = \frac{220 - 7(27)}{12} = \frac{31}{12} \quad \text{(not a natural number)} \]

28. **For \( x = 28 \):**
\[ y = \frac{220 - 7(28)}{12} = \frac{24}{12} = 2 \quad \text{(natural number)} \]

29. **For \( x = 29 \):**
\[ y = \frac{220 - 7(29)}{12} = \frac{17}{12} \quad \text{(not a natural number)} \]

30. **For \( x = 30 \):**
\[ y = \frac{220 - 7(30)}{12} = \frac{10}{12} \quad \text{(not a natural number)} \]

31. **For \( x = 31 \):**
\[ y = \frac{220 - 7(31)}{12} = \frac{3}{12} \quad \text{(not a natural number)} \]

### Step 4: Summary of Solutions
The valid pairs of \( (x, y) \) that are natural numbers are:
- \( (4, 16) \)
- \( (16, 9) \)
- \( (28, 2) \)

Thus, there are **3 solutions** in total.

### Final Answer
The number of solutions is **3**.

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