On selling a television at a $6\%$ profit and a fridge at a $9\%$ profit, a shopkeeper gains $\text{Rs. }5,100$. However, if the shopkeeper sells the television at a $9\%$ profit and the fridge at a $6\%$ loss, he gains $\text{Rs. }1,800$ on the whole transaction. The cost price (in $\text{Rs.}$) of the fridge is:

Let the cost price of the television be $T$ and the cost price of the fridge be $F$.From the first condition:
$$0.06T+0.09F=5100⟹6T+9F=510000⟹2T+3F=170000\text{— (Equation 1)}$$
From the second condition:
$$0.09T-0.06F=1800⟹9T-6F=180000⟹3T-2F=60000\text{— (Equation 2)}$$
To find the cost price of the fridge ($F$), we eliminate $T$:Multiply Equation 1 by $3$:
$$6T+9F=510000$$
Multiply Equation 2 by $2$:
$$6T-4F=120000$$
Subtract the second derived equation from the first:
$$(6T+9F)-(6T-4F)=510000-120000$$
$$13F=390000$$
$$F=30000$$
Thus, the cost price of the fridge is $\text{Rs. }30,000$.