1. **State the problem:** Solve the simultaneous equations: $$-x + 3y = 7$$ $$2x^2 + 3y^2 = 35$$ 2. **Express one variable in terms of the other:** From the first equation, $$-x + 3y = 7 \implies x = 3y - 7$$ 3. **Substitute into the second equation:** Replace $x$ with $3y - 7$ in the second equation: $$2(3y - 7)^2 + 3y^2 = 35$$ 4. **Expand and simplify:** $$2(9y^2 - 42y + 49) + 3y^2 = 35$$ $$18y^2 - 84y + 98 + 3y^2 = 35$$ $$21y^2 - 84y + 98 = 35$$ 5. **Bring all terms to one side:** $$21y^2 - 84y + 98 - 35 = 0$$ $$21y^2 - 84y + 63 = 0$$ 6. **Divide entire equation by 21 to simplify:** $$\cancel{21}y^2 - \cancel{21}4y + \cancel{21}3 = 0 \implies y^2 - 4y + 3 = 0$$ 7. **Factor the quadratic:** $$(y - 3)(y - 1) = 0$$ 8. **Solve for $y$:** $$y = 3 \quad \text{or} \quad y = 1$$ 9. **Find corresponding $x$ values:** For $y = 3$: $$x = 3(3) - 7 = 9 - 7 = 2$$ For $y = 1$: $$x = 3(1) - 7 = 3 - 7 = -4$$ 10. **Final solutions:** $$(x, y) = (2, 3) \quad \text{or} \quad (-4, 1)$$