To solve the equation $\frac{1}{3} \left(\frac{3}{4}x + 5\right) - 3 = a \left(\frac{1}{2}x + b\right)$, we can simplify and analyze it step by step.

---

Step 1: Distribute and simplify the left-hand side.

$\frac{1}{3} \left(\frac{3}{4}x + 5\right) = \frac{1}{3} \cdot \frac{3}{4}x + \frac{1}{3} \cdot 5 = \frac{1}{4}x + \frac{5}{3}$ Thus, the left-hand side becomes: $\frac{1}{4}x + \frac{5}{3} - 3$ Combine the constants: $\frac{5}{3} - 3 = \frac{5}{3} - \frac{9}{3} = -\frac{4}{3}$ So the left-hand side simplifies to: $\frac{1}{4}x - \frac{4}{3}$

---

Step 2: Expand the right-hand side.

$a \left(\frac{1}{2}x + b\right) = a \cdot \frac{1}{2}x + a \cdot b = \frac{a}{2}x + ab$

---

Step 3: Combine both sides.

Equating the two sides: $\frac{1}{4}x - \frac{4}{3} = \frac{a}{2}x + ab$

---

Step 4: Separate coefficients of $x$ and constants.

1. Coefficient of $x$: $\frac{1}{4} = \frac{a}{2}$ Solve for $a$: $a = \frac{1}{4} \cdot 2 = \frac{1}{2}$

2. Constant terms: $-\frac{4}{3} = ab$ Substitute $a = \frac{1}{2}$: $-\frac{4}{3} = \frac{1}{2}b$ Solve for $b$: $b = -\frac{4}{3} \cdot 2 = -\frac{8}{3}$

---

Final Answer:

$a = \frac{1}{2}, \, b = -\frac{8}{3}$

Would you like further clarification or additional details?

---

Related Questions:

1. How do you verify solutions for equations with variables and fractions?
2. Can this be solved graphically, and if so, how would the graphs compare?
3. What are the general steps for solving linear equations with multiple terms?
4. How do coefficients affect the behavior of linear functions?
5. How can similar equations be solved using matrix methods?

---

Tip:

Always simplify fractions and constants first to make solving equations easier!