Solve System Ff9A0A

1. **State the problem:** We are given two equations with two variables $x$ and $y$: $$7(x+y) - 9(x-y) = 3(x+y)^2$$ $$5(x+y) - 8(x-y) = (x+y)^2$$ 2. **Rewrite the equations:** Let $a = x+y$ and $b = x-y$ to simplify the system. The system becomes: $$7a - 9b = 3a^2$$ $$5a - 8b = a^2$$ 3. **Express $b$ from the second equation:** $$5a - 8b = a^2 \implies -8b = a^2 - 5a \implies b = \frac{5a - a^2}{8}$$ 4. **Substitute $b$ into the first equation:** $$7a - 9\left(\frac{5a - a^2}{8}\right) = 3a^2$$ Multiply both sides by 8 to clear the denominator: $$8 \times 7a - 9(5a - a^2) = 8 \times 3a^2$$ $$56a - 45a + 9a^2 = 24a^2$$ Simplify: $$11a + 9a^2 = 24a^2$$ 5. **Bring all terms to one side:** $$11a + 9a^2 - 24a^2 = 0 \implies 11a - 15a^2 = 0$$ 6. **Factor the equation:** $$a(11 - 15a) = 0$$ So, $$a = 0 \quad \text{or} \quad 11 - 15a = 0 \implies a = \frac{11}{15}$$ 7. **Find corresponding $b$ values:** - For $a=0$: $$b = \frac{5 \times 0 - 0^2}{8} = 0$$ - For $a=\frac{11}{15}$: $$b = \frac{5 \times \frac{11}{15} - \left(\frac{11}{15}\right)^2}{8} = \frac{\frac{55}{15} - \frac{121}{225}}{8} = \frac{\frac{825}{225} - \frac{121}{225}}{8} = \frac{\frac{704}{225}}{8} = \frac{704}{1800} = \frac{88}{225}$$ 8. **Recall $a = x+y$ and $b = x-y$, solve for $x$ and $y$:** $$x = \frac{a + b}{2}, \quad y = \frac{a - b}{2}$$ - For $a=0$, $b=0$: $$x = \frac{0+0}{2} = 0, \quad y = \frac{0-0}{2} = 0$$ - For $a=\frac{11}{15}$, $b=\frac{88}{225}$: $$x = \frac{\frac{11}{15} + \frac{88}{225}}{2} = \frac{\frac{165}{225} + \frac{88}{225}}{2} = \frac{\frac{253}{225}}{2} = \frac{253}{450}$$ $$y = \frac{\frac{11}{15} - \frac{88}{225}}{2} = \frac{\frac{165}{225} - \frac{88}{225}}{2} = \frac{\frac{77}{225}}{2} = \frac{77}{450}$$ **Final solutions:** $$\boxed{(x,y) = (0,0) \quad \text{or} \quad \left(\frac{253}{450}, \frac{77}{450}\right)}$$