1. **State the problem:** Solve the simultaneous equations: $$7x + 11y = -5$$ $$4x + 3y = 7$$ 2. **Choose a method:** We will use the elimination method to solve for $x$ and $y$. 3. **Eliminate one variable:** Multiply the first equation by 3 and the second equation by 11 to align coefficients of $y$: $$3(7x + 11y) = 3(-5) \Rightarrow 21x + 33y = -15$$ $$11(4x + 3y) = 11(7) \Rightarrow 44x + 33y = 77$$ 4. **Subtract the first new equation from the second:** $$\cancel{33y} + 44x - (21x + \cancel{33y}) = 77 - (-15)$$ $$44x - 21x = 77 + 15$$ $$23x = 92$$ 5. **Solve for $x$:** $$x = \frac{92}{23} = 4$$ 6. **Substitute $x=4$ into one original equation to find $y$:** Using $4x + 3y = 7$: $$4(4) + 3y = 7$$ $$16 + 3y = 7$$ 7. **Isolate $y$:** $$3y = 7 - 16$$ $$3y = -9$$ 8. **Solve for $y$:** $$y = \frac{-9}{3} = -3$$ **Final answer:** $$x = 4, \quad y = -3$$