Question: User: 
1. **Problem Statement:**
Find the value of $x$ that satisfies the equation $$\frac{2x+3}{5} = \frac{3x-1}{4}.$$
2. **Formula and Rules:**
To solve an equation with fractions, we can cross-multiply to eliminate the denominators. The rule is:
$$\frac{a}{b} = \frac{c}{d} \implies ad = bc.$$
3. **Cross-multiply:**
Multiply both sides by $5 \times 4 = 20$ or directly cross-multiply:
$$4(2x + 3) = 5(3x - 1).$$
4. **Expand both sides:**
$$8x + 12 = 15x - 5.$$
5. **Isolate $x$ terms on one side:**
Subtract $8x$ from both sides:
$$\cancel{8x} + 12 = 15x - 5 - \cancel{8x} \implies 12 = 7x - 5.$$
6. **Isolate constant terms:**
Add $5$ to both sides:
$$12 + 5 = 7x - 5 + 5 \implies 17 = 7x.$$
7. **Solve for $x$:**
Divide both sides by $7$:
$$\frac{17}{\cancel{7}} = x \frac{\cancel{7}}{7} \implies x = \frac{17}{7}.$$
8. **Final answer:**
$$\boxed{x = \frac{17}{7}}.$$