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Fractions can seem confusing, but they become simple with practice. A fraction consists of a numerator (top number) and a denominator (bottom number).
To add or subtract fractions, their denominators must be the same. If they're the same, you simply add or subtract the numerators and keep the denominator.
For example, adding \(\frac{5}{12}\) and \(-\frac{7}{12}\) means summing the numerators while keeping the denominator 12: \(\frac{5 - 7}{12} = -\frac{2}{12}\). We often simplify fractions by dividing both the numerator and denominator by their greatest common factor (GCF).

In our problem, we dealt with fractions directly without needing to simplify initially. Recognizing that \(\frac{-12}{12} = -1\) comes from understanding that any fraction with identical numerator and denominator equals 1.

Isolating the variable means having the variable \(z\) by itself on one side of the equation. This is a key step in solving linear equations.
In our problem, after simplifying the left-hand side, we ended up with \(\frac{5}{12}\) added to \(z\).
To isolate \(z\), we subtracted \(\frac{5}{12}\) from both sides. This keeps the equation balanced. Remember, whatever you do to one side, you must do to the other!

Ensuring \(z\) Is isolated lets us find its exact value. Isolation transforms the equation into \(z = \text{value},\) making math easier to manage and understand.

Simplification of equations involves making the expressions as straightforward as possible. In our initial step, recognizing that subtracting a negative number can be turned into adding a positive number simplified our work.
Simplifying \(-\frac{12}{12}\) to \(-1\) ensures our answers are clear and understandable. Always seek to break down equations into their simplest forms.

Simplification also involves combining like terms. For instance, when we added fractions, \(-\frac{7}{12} - \frac{5}{12}\), we combined them into a single fraction. Having clear and simplified results prevents confusion and mistakes.
Understanding each step deeply ensures confidence in solving future problems and further applications in algebra.