1. The problem is to verify if the algebraic rearrangements and formulas given for various expressions involving variables $m$, $n$, $p$, $x$, and $y$ are correct. 2. The general rule for solving for a variable in an equation is to isolate that variable on one side using inverse operations such as addition/subtraction and multiplication/division. 3. Let's check the first expression as an example: Given: $6 - 2m - n$ with $[m]$ and the formula $\frac{n + 6}{2} = m$ Starting from $6 - 2m - n$, rearranged to solve for $m$: $$6 - n = 2m$$ Divide both sides by 2: $$m = \frac{6 - n}{2}$$ The given formula is $m = \frac{n + 6}{2}$ which is different from $\frac{6 - n}{2}$. So the given formula is incorrect for this expression. 4. Checking the third expression: Given: $10 = 5m - n$ with $[m]$ and formula $\frac{n + 10}{5} = m$ Rearranged: $$5m = 10 + n$$ $$m = \frac{10 + n}{5}$$ This matches the given formula, so it is correct. 5. Checking the fourth expression: Given: $6 = 2m - n$ with $[n]$ and formula $n = 2m + 6$ Rearranged: $$n = 2m - 6$$ The given formula is $n = 2m + 6$, which is incorrect. 6. The pattern is that some formulas are correct, others have sign errors or incorrect rearrangements. 7. To verify correctness, always isolate the variable step-by-step and check if the given formula matches the algebraic manipulation. Final conclusion: Some formulas are correct, others are not. Careful sign and operation checks are needed for each.