Question: Solve each equation.
$$\frac{6x + 7}{7} - 3 = 5x - \frac{3}{8}$$
1. **State the problem:** Solve the equation $$\frac{6x + 7}{7} - 3 = 5x - \frac{3}{8}$$ for $$x$$.
2. **Rewrite the equation:**
$$\frac{6x + 7}{7} - 3 = 5x - \frac{3}{8}$$
3. **Isolate terms involving $$x$$:**
Add $$3$$ to both sides:
$$\frac{6x + 7}{7} = 5x - \frac{3}{8} + 3$$
4. **Convert $$3$$ to a fraction with denominator $$8$$:**
$$3 = \frac{24}{8}$$
So,
$$5x - \frac{3}{8} + 3 = 5x - \frac{3}{8} + \frac{24}{8} = 5x + \frac{21}{8}$$
5. **Rewrite the equation:**
$$\frac{6x + 7}{7} = 5x + \frac{21}{8}$$
6. **Clear denominators by multiplying both sides by the least common multiple of $$7$$ and $$8$$, which is $$56$$:**
$$56 \times \frac{6x + 7}{7} = 56 \times \left(5x + \frac{21}{8}\right)$$
7. **Simplify each side:**
$$56 \times \frac{6x + 7}{7} = 8(6x + 7) = 48x + 56$$
$$56 \times 5x = 280x$$
$$56 \times \frac{21}{8} = 7 \times 21 = 147$$
So,
$$48x + 56 = 280x + 147$$
8. **Bring all $$x$$ terms to one side and constants to the other:**
$$48x - 280x = 147 - 56$$
$$-232x = 91$$
9. **Solve for $$x$$:**
$$x = \frac{91}{-232} = -\frac{91}{232}$$
10. **Simplify the fraction if possible:**
Both numerator and denominator share a common factor of $$1$$ only, so the fraction is already in simplest form.
**Final answer:**
$$x = -\frac{91}{232}$$