Relation \( 2a + 3b = 30 \) on \( \mathbb{N} \)
📺 Video Explanation
📝 Question
Let relation \( R \) on \( \mathbb{N} \) be defined as:
\[ aRb \iff 2a + 3b = 30 \]
Write \( R \) as a set of ordered pairs.
✅ Solution
🔹 Step 1: Express a in terms of b
\[ 2a + 3b = 30 \Rightarrow a = \frac{30 – 3b}{2} \]
🔹 Step 2: Find Natural Number Values
For \( a \) to be natural, \( (30 – 3b) \) must be even.
So, \( 3b \) must be even ⇒ \( b \) must be even.
Try even values of \( b \):
- \( b = 2 \Rightarrow a = 12 \)
- \( b = 4 \Rightarrow a = 9 \)
- \( b = 6 \Rightarrow a = 6 \)
- \( b = 8 \Rightarrow a = 3 \)
- \( b = 10 \Rightarrow a = 0 \) (not natural)
🔹 Step 3: Write Ordered Pairs
\[ (12,2), (9,4), (6,6), (3,8) \]
🎯 Final Answer
\[ \boxed{R = \{(12,2), (9,4), (6,6), (3,8)\}} \]
🚀 Exam Insight
- Convert equation to one variable form
- Check divisibility carefully
- Ensure values belong to \( \mathbb{N} \)