Since \( y \) varies directly as \( x \), we can express their relationship with the equation:
\[ y = kx \]
where \( k \) is the constant of variation.
From the information given, when \( y = 10 \), \( x = 6 \). We can use this to find \( k \):
\[ 10 = k \cdot 6 \]
Solving for \( k \):
\[ k = \frac{10}{6} = \frac{5}{3} \]
Now that we have the constant \( k \), we can find \( x \) when \( y = 14 \):
\[ 14 = \frac{5}{3} x \]
To isolate \( x \), multiply both sides by \( \frac{3}{5} \):
\[ x = 14 \cdot \frac{3}{5} = \frac{42}{5} = 8.4 \]
Therefore, when \( y = 14 \), the value of \( x \) is
\[ \boxed{8.4} \]