1. **State the problem:** Given the equation $$2x + 3y = 4 \times \frac{1}{8}$$, show that $$5x + 2y = -3$$. 2. **Simplify the given equation:** Calculate the right side: $$4 \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2}$$ So the equation becomes: $$2x + 3y = \frac{1}{2}$$ 3. **Express one variable in terms of the other:** Let's solve for $$x$$: $$2x = \frac{1}{2} - 3y$$ $$x = \frac{\frac{1}{2} - 3y}{2} = \frac{1}{4} - \frac{3y}{2}$$ 4. **Substitute $$x$$ into the equation to prove:** We want to show: $$5x + 2y = -3$$ Substitute $$x = \frac{1}{4} - \frac{3y}{2}$$: $$5\left(\frac{1}{4} - \frac{3y}{2}\right) + 2y = -3$$ 5. **Expand and simplify:** $$5 \times \frac{1}{4} - 5 \times \frac{3y}{2} + 2y = -3$$ $$\frac{5}{4} - \frac{15y}{2} + 2y = -3$$ 6. **Combine like terms:** Convert $$2y$$ to $$\frac{4y}{2}$$ to combine with $$-\frac{15y}{2}$$: $$\frac{5}{4} - \frac{15y}{2} + \frac{4y}{2} = -3$$ $$\frac{5}{4} - \frac{11y}{2} = -3$$ 7. **Isolate $$y$$:** Subtract $$\frac{5}{4}$$ from both sides: $$- \frac{11y}{2} = -3 - \frac{5}{4}$$ Find common denominator for right side: $$-3 = -\frac{12}{4}$$ So: $$- \frac{11y}{2} = -\frac{12}{4} - \frac{5}{4} = -\frac{17}{4}$$ 8. **Divide both sides by $$-\frac{11}{2}$$:** $$y = \frac{-\frac{17}{4}}{-\frac{11}{2}} = \frac{-17}{4} \times \frac{2}{-11}$$ Cancel negatives: $$y = \frac{17}{4} \times \frac{2}{11}$$ Multiply numerators and denominators: $$y = \frac{34}{44}$$ Simplify fraction: $$y = \frac{17}{22}$$ 9. **Find $$x$$ using $$y$$:** Recall: $$x = \frac{1}{4} - \frac{3y}{2}$$ Substitute $$y = \frac{17}{22}$$: $$x = \frac{1}{4} - \frac{3}{2} \times \frac{17}{22} = \frac{1}{4} - \frac{51}{44}$$ Convert $$\frac{1}{4}$$ to $$\frac{11}{44}$$: $$x = \frac{11}{44} - \frac{51}{44} = -\frac{40}{44} = -\frac{10}{11}$$ 10. **Verify the equation $$5x + 2y = -3$$:** Substitute $$x = -\frac{10}{11}$$ and $$y = \frac{17}{22}$$: $$5 \times -\frac{10}{11} + 2 \times \frac{17}{22} = -\frac{50}{11} + \frac{34}{22}$$ Convert $$-\frac{50}{11}$$ to $$-\frac{100}{22}$$: $$-\frac{100}{22} + \frac{34}{22} = -\frac{66}{22} = -3$$ **Final answer:** $$5x + 2y = -3$$ is true given $$2x + 3y = 4 \times \frac{1}{8}$$.