The total price of $6$ shirts and $7$ trousers is $\text{Rs. }4,130$ while that of $4$ shirts and $9$ trousers is $\text{Rs. }4,270$. Find out the price of $3$ shirts.

Let the price of one shirt be $S$ and one pair of trousers be $T$.
From the given conditions:

$$6S+7T=4130\text{— (Equation 1)}$$

$$4S+9T=4270\text{— (Equation 2)}$$

Multiply Equation 1 by $2$ and Equation 2 by $3$:

$$12S+14T=8260\text{— (Equation 3)}$$

$$12S+27T=12810\text{— (Equation 4)}$$

Subtract Equation 3 from Equation 4:

$$(27T-14T)=12810-8260$$

$$13T=4550$$

$$T=\frac{4550}{13}=350$$

Substitute $T=350$ back into Equation 1:

$$6S+7(350)=4130$$

$$6S+2450=4130$$

$$6S=4130-2450=1680$$

$$S=\frac{1680}{6}=280$$

The cost of $3$ shirts is:

$$3S=3\times 280=\text{Rs. }840$$