1. **State the problem:** Solve the simultaneous equations: $$4x - 5y = 20$$ $$6x + 7y = -57$$ 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 7 and the second by 5 to align coefficients of $y$: $$7(4x - 5y) = 7(20) \Rightarrow 28x - 35y = 140$$ $$5(6x + 7y) = 5(-57) \Rightarrow 30x + 35y = -285$$ 4. **Add the two equations to eliminate $y$:** $$28x - 35y + 30x + 35y = 140 - 285$$ $$\Rightarrow (28x + 30x) + (-35y + 35y) = -145$$ $$\Rightarrow 58x + \cancel{-35y + 35y} = -145$$ 5. **Simplify and solve for $x$:** $$58x = -145$$ $$x = \frac{-145}{58}$$ $$x = -\frac{145}{58}$$ 6. **Substitute $x$ back into the first equation to solve for $y$:** $$4x - 5y = 20$$ $$4\left(-\frac{145}{58}\right) - 5y = 20$$ $$-\frac{580}{58} - 5y = 20$$ $$-10 - 5y = 20$$ 7. **Isolate $y$:** $$-5y = 20 + 10$$ $$-5y = 30$$ $$y = \frac{30}{-5}$$ $$y = -6$$ **Final answer:** $$x = -\frac{145}{58}, \quad y = -6$$