1. **State the problem:** We are given a system of equations: $$x - xy + y = -7$$ $$xy = 15$$ We need to find the value of $$x_1^2 + y_1^2$$ where $$(x_1, y_1)$$ is a solution to the system. 2. **Rewrite the first equation:** $$x - xy + y = -7$$ Group terms: $$x + y - xy = -7$$ 3. **Use the second equation:** $$xy = 15$$ 4. **Express the sum $$x + y$$:** From the first equation: $$x + y - xy = -7 \implies x + y = -7 + xy$$ Substitute $$xy = 15$$: $$x + y = -7 + 15 = 8$$ 5. **Recall the identity for sum of squares:** $$x^2 + y^2 = (x + y)^2 - 2xy$$ 6. **Calculate $$x_1^2 + y_1^2$$:** $$x_1^2 + y_1^2 = 8^2 - 2 \times 15 = 64 - 30 = 34$$ **Final answer:** $$\boxed{34}$$