1. The problem is to find the missing value of $x$ in the ordered pair $(x, -6)$ that satisfies the equation $4x + y = 10$. 2. Substitute $y = -6$ into the equation: $4x + (-6) = 10$. 3. Simplify the equation: $4x - 6 = 10$. 4. Add 6 to both sides to isolate the term with $x$: $4x = 10 + 6$. 5. Calculate the right side: $4x = 16$. 6. Divide both sides by 4 to solve for $x$: $x = \frac{16}{4}$. 7. Simplify the fraction: $x = 4$. 8. Therefore, the missing value is $4$, and the ordered pair is $(4, -6)$.