Question: top-left: $\frac{4}{5} x + \frac{6}{7} = 7x + \frac{87}{4}$
1. **State the problem:** Solve the equation $$\frac{4}{5} x + \frac{6}{7} = 7x + \frac{87}{4}$$ for $x$.
2. **Write down the equation:** $$\frac{4}{5} x + \frac{6}{7} = 7x + \frac{87}{4}$$
3. **Goal:** Isolate $x$ on one side.
4. **Subtract $7x$ from both sides:**
$$\frac{4}{5} x - 7x + \frac{6}{7} = \frac{87}{4}$$
5. **Combine like terms for $x$:**
$$\frac{4}{5} x - 7x = \frac{4}{5} x - \frac{35}{5} x = \frac{4 - 35}{5} x = \frac{-31}{5} x$$
So the equation becomes:
$$\frac{-31}{5} x + \frac{6}{7} = \frac{87}{4}$$
6. **Subtract $\frac{6}{7}$ from both sides:**
$$\frac{-31}{5} x = \frac{87}{4} - \frac{6}{7}$$
7. **Find common denominator for right side:**
The denominators are $4$ and $7$, common denominator is $28$.
$$\frac{87}{4} = \frac{87 \times 7}{28} = \frac{609}{28}$$
$$\frac{6}{7} = \frac{6 \times 4}{28} = \frac{24}{28}$$
So,
$$\frac{87}{4} - \frac{6}{7} = \frac{609}{28} - \frac{24}{28} = \frac{585}{28}$$
8. **Rewrite equation:**
$$\frac{-31}{5} x = \frac{585}{28}$$
9. **Solve for $x$ by dividing both sides by $\frac{-31}{5}$:**
$$x = \frac{585}{28} \div \frac{-31}{5} = \frac{585}{28} \times \frac{5}{-31}$$
10. **Multiply numerators and denominators:**
$$x = \frac{585 \times 5}{28 \times (-31)} = \frac{2925}{-868}$$
11. **Simplify fraction:**
Find gcd of 2925 and 868.
- $2925 = 25 \times 117$
- $868 = 4 \times 217$
No common factors other than 1, so fraction is simplified.
12. **Final answer:**
$$x = -\frac{2925}{868}$$