Solve the equation 5(-5 + n) - (-5n + 1) = -2 and give the answer as a fraction.

Step-by-step solution

1. 1

   Expand the first bracket.

   $5(-5 + n) = -25 + 5n$

2. 2

   Distribute the minus over the second bracket. The leading sign flips both terms inside:

   $-(-5n + 1) = +5n - 1$

   Getting $-5n - 1$ here instead would make the $n$ terms cancel and produce a false "no solution", which is the classic failure mode for this problem.

3. 3

   Combine like terms.

   $-25 + 5n + 5n - 1 = -2 \quad\Longrightarrow\quad 10n - 26 = -2$

4. 4

   Isolate n.

   $10n = 24 \quad\Longrightarrow\quad n = \frac{24}{10} = \frac{12}{5} = 2.4$

   Reducing $\tfrac{24}{10}$ by the common factor $2$ gives the fraction in lowest terms.

5. 5

   Check by substitution. $5\left(-5 + 2.4\right) = 5(-2.6) = -13$, and $-\left(-5(2.4) + 1\right) = -(-12 + 1) = 11$. Their sum is $-13 + 11 = -2$, matching the right-hand side.