The determinant of a 2x2 matrix \(\begin{vmatrix} a & b \\ c & d \end{vmatrix}\) is calculated as \(ad - bc\).

In this case, you are given that \(\begin{vmatrix} x & 3 \\ 2 & 7 \end{vmatrix} = 15\).

So, the equation becomes:

\(x \cdot 7 - 3 \cdot 2 = 15\)

Simplify:

7x - 6 = 15

Add 6 to both sides:

7x = 21

Divide by 7:

x = 3

So, the value of x is 3.