To solve the equation
\[ \frac{x-5}{6} = \frac{x+2}{7}, \]
we can use cross-multiplication. This results in:
\[ 7(x - 5) = 6(x + 2). \]
Now, let's expand both sides:
\[ 7x - 35 = 6x + 12. \]
Next, we can isolate \(x\) by first subtracting \(6x\) from both sides:
\[ 7x - 6x - 35 = 12 \ x - 35 = 12. \]
Now, add \(35\) to both sides:
\[ x = 12 + 35 \ x = 47. \]
Thus, the solution to the equation is
\[ \boxed{47}. \]