Find the Linear Relation Between the Components of Ordered Pairs of R
Question
If
\[ R=\{(2,1),(4,7),(1,-2),\ldots\}, \]
then write the linear relation between the components of the ordered pairs of the relation \( R \).
Solution
Let the linear relation be
\[ y=ax+b \]
Using the pair \( (2,1) \),
\[ 1=2a+b \]
Using the pair \( (4,7) \),
\[ 7=4a+b \]
Subtracting,
\[ 6=2a \]
\[ a=3 \]
Putting in
\[ 1=2a+b \]
\[ 1=2(3)+b \]
\[ b=-5 \]
Therefore,
\[ y=3x-5 \]
Checking for \( (1,-2) \),
\[ 3(1)-5=-2 \]
Hence, the required linear relation is:
\[ \boxed{y=3x-5} \]