X Y Difference 20Fb94

1. **State the problem:** We have two equations: $$6x + b = x - 2$$ $$6y + c = y - 2$$ where $b$ and $c$ are constants, and $b$ is $\frac{1}{2}$ less than $c$, i.e., $$b = c - \frac{1}{2}$$ We want to find the relationship between $x$ and $y$. 2. **Rewrite each equation to solve for $x$ and $y$: ** From the first equation: $$6x + b = x - 2$$ Subtract $x$ from both sides: $$6x - x + b = -2$$ $$5x + b = -2$$ Subtract $b$ from both sides: $$5x = -2 - b$$ Divide both sides by 5: $$x = \frac{-2 - b}{5}$$ 3. Similarly, from the second equation: $$6y + c = y - 2$$ Subtract $y$ from both sides: $$6y - y + c = -2$$ $$5y + c = -2$$ Subtract $c$ from both sides: $$5y = -2 - c$$ Divide both sides by 5: $$y = \frac{-2 - c}{5}$$ 4. **Express $x - y$ to find their difference:** $$x - y = \frac{-2 - b}{5} - \frac{-2 - c}{5} = \frac{-2 - b + 2 + c}{5} = \frac{c - b}{5}$$ 5. **Use the relation between $b$ and $c$: ** Given: $$b = c - \frac{1}{2}$$ So, $$c - b = c - (c - \frac{1}{2}) = \frac{1}{2}$$ 6. **Substitute back:** $$x - y = \frac{1/2}{5} = \frac{1}{10}$$ 7. **Interpretation:** $x$ is $\frac{1}{10}$ greater than $y$. **Answer: (B) x is 1/10 greater than y**