Question: Solve for $z$. Express your answer as a proper or improper fraction in simplest terms.
$\frac{5}{12} z + \frac{1}{4} = -\frac{1}{4}$
1. **State the problem:** Solve for $z$ in the equation $$\frac{5}{12} z + \frac{1}{4} = -\frac{1}{4}$$.
2. **Isolate the term with $z$:** Subtract $\frac{1}{4}$ from both sides:
$$\frac{5}{12} z + \frac{1}{4} - \frac{1}{4} = -\frac{1}{4} - \frac{1}{4}$$
which simplifies to
$$\frac{5}{12} z = -\frac{2}{4}$$
3. **Simplify the right side:**
$$-\frac{2}{4} = -\frac{1}{2}$$
So now we have
$$\frac{5}{12} z = -\frac{1}{2}$$
4. **Solve for $z$ by dividing both sides by $\frac{5}{12}$:**
$$z = \frac{-\frac{1}{2}}{\frac{5}{12}}$$
Rewrite division as multiplication by reciprocal:
$$z = -\frac{1}{2} \times \frac{12}{5}$$
5. **Multiply numerators and denominators:**
$$z = -\frac{1 \times 12}{2 \times 5} = -\frac{12}{10}$$
6. **Simplify the fraction by dividing numerator and denominator by their greatest common divisor 2:**
$$z = -\frac{\cancel{12}^{6}}{\cancel{10}^{5}} = -\frac{6}{5}$$
**Final answer:**
$$z = -\frac{6}{5}$$