1. **State the problem:** Simplify the given expression step-by-step: $$\{(x,y) : x^2 - y^2\} + \{(3, 9+3) + (3, \frac{8}{3} \div \frac{1}{5})\} - 2 \{(1) \times (-1) - 6\} \div 4 \times 3 - 6$$ and then simplify the sequence of sums and differences: $$\{(-7) + (-6) + (-8) + (-7) + (-3)\} = \{(-3) - 2 \times (-7) + (-3) + (-1) - 9\} + \{(-3)7\} = ...$$ 2. **Simplify each part:** - For $\{(x,y) : x^2 - y^2\}$, this is a set definition, no numeric simplification. - Calculate $9+3=12$. - Calculate $\frac{8}{3} \div \frac{1}{5} = \frac{8}{3} \times 5 = \frac{40}{3}$. - So $\{(3, 12) + (3, \frac{40}{3})\} = \{(3+3, 12 + \frac{40}{3})\} = \{(6, \frac{36}{3} + \frac{40}{3})\} = \{(6, \frac{76}{3})\}$. - Next, $\{(1) \times (-1) - 6\} = -1 - 6 = -7$. - Then $-2 \times (-7) \div 4 \times 3 - 6$: First, $-2 \times (-7) = 14$. Then $14 \div 4 = 3.5$. Then $3.5 \times 3 = 10.5$. Finally, $10.5 - 6 = 4.5$. 3. **Simplify the sums:** - $\{(-7) + (-6) + (-8) + (-7) + (-3)\} = -7 -6 -8 -7 -3 = -31$. - $\{(-3) - 2 \times (-7) + (-3) + (-1) - 9\} + \{(-3)7\}$: Calculate $-2 \times (-7) = 14$. So inside the first bracket: $-3 + 14 - 3 - 1 - 9 = (-3 - 3 - 1 - 9) + 14 = (-16) + 14 = -2$. The second bracket: $(-3)7 = -21$. Sum: $-2 + (-21) = -23$. - The next lines seem inconsistent or have errors, but following the last correct step: $(-3) - 7 - (-3) - (-1) - (-3) + 3 = (-3 - 7 + 3 + 1 + 3) + 3 = (-7) + 3 = -4$ (rechecking carefully: $-3 -7 = -10$, $-10 - (-3) = -10 + 3 = -7$, $-7 - (-1) = -7 + 1 = -6$, $-6 - (-3) = -6 + 3 = -3$, then $-3 + 3 = 0$) So the sum is $0$. - Then $(-7) - (-3) - (-1) - (-3) + 3 = (-7 + 3 + 1 + 3) + 3 = 0 + 3 = 3$. - $ (2) - 7 = -5$. - $222 + (-17) + (-3) = 222 - 17 - 3 = 202$. - $33 - (-1) - (-3) = 33 + 1 + 3 = 37$. - $96 - (-3) - (-1) = 96 + 3 + 1 = 100$. - $18 - 36 = -18$. 4. **Final answers:** - The simplified vector sum is $\{(6, \frac{76}{3})\}$. - The arithmetic expressions simplify to: - $-31$ - $-23$ - $0$ - $3$ - $-5$ - $202$ - $37$ - $100$ - $-18$