Question: top-left: $\frac{1}{2}(-8x + 7) = -x - \frac{4}{3}(-9x + 3)$
1. **State the problem:** Solve the equation $$\frac{1}{2}(-8x + 7) = -x - \frac{4}{3}(-9x + 3)$$ for $x$.
2. **Write the equation clearly:**
$$\frac{1}{2}(-8x + 7) = -x - \frac{4}{3}(-9x + 3)$$
3. **Distribute the fractions:**
$$\frac{1}{2} \times -8x + \frac{1}{2} \times 7 = -x - \frac{4}{3} \times -9x - \frac{4}{3} \times 3$$
4. **Calculate each term:**
$$-4x + \frac{7}{2} = -x + 12x - 4$$
5. **Simplify the right side:**
$$-4x + \frac{7}{2} = 11x - 4$$
6. **Bring all $x$ terms to one side and constants to the other:**
$$-4x - 11x = -4 - \frac{7}{2}$$
7. **Combine like terms:**
$$-15x = -4 - \frac{7}{2}$$
8. **Convert $-4$ to fraction with denominator 2:**
$$-4 = -\frac{8}{2}$$
9. **Add the constants:**
$$-15x = -\frac{8}{2} - \frac{7}{2} = -\frac{15}{2}$$
10. **Divide both sides by $-15$ to solve for $x$:**
$$x = \frac{-\frac{15}{2}}{-15}$$
11. **Simplify the division:**
$$x = \frac{-15}{2} \times \frac{1}{-15} = \frac{\cancel{-15}}{2} \times \frac{1}{\cancel{-15}} = \frac{1}{2}$$
**Final answer:**
$$x = \frac{1}{2}$$