1. **State the problem:** We are given the vector equation $$5\begin{pmatrix}7\\-\frac{1}{4}x\\12\end{pmatrix} = \frac{9}{2}\begin{pmatrix}y\\y\\y\end{pmatrix}$$ and need to find the values of $x$ and $y$. 2. **Write the equation component-wise:** Multiply the scalar 5 on the left vector: $$\begin{pmatrix}5 \times 7\\5 \times -\frac{1}{4}x\\5 \times 12\end{pmatrix} = \begin{pmatrix}\frac{9}{2}y\\\frac{9}{2}y\\\frac{9}{2}y\end{pmatrix}$$ which simplifies to $$\begin{pmatrix}35\\-\frac{5}{4}x\\60\end{pmatrix} = \begin{pmatrix}\frac{9}{2}y\\\frac{9}{2}y\\\frac{9}{2}y\end{pmatrix}$$ 3. **Set corresponding components equal:** - First component: $$35 = \frac{9}{2}y$$ - Second component: $$-\frac{5}{4}x = \frac{9}{2}y$$ - Third component: $$60 = \frac{9}{2}y$$ 4. **Solve for $y$ using the first component:** $$35 = \frac{9}{2}y$$ Multiply both sides by the reciprocal of $\frac{9}{2}$: $$y = 35 \times \frac{2}{9} = \frac{70}{9}$$ 5. **Check $y$ with the third component:** $$60 = \frac{9}{2}y$$ Substitute $y = \frac{70}{9}$: $$60 = \frac{9}{2} \times \frac{70}{9} = \frac{70}{2} = 35$$ This is not true, so there is a contradiction. 6. **Check for consistency:** Since the first and third components give different values for $\frac{9}{2}y$ (35 and 60), the system is inconsistent unless $y$ satisfies both. 7. **Solve for $y$ using the third component:** $$60 = \frac{9}{2}y$$ Multiply both sides by $\frac{2}{9}$: $$y = 60 \times \frac{2}{9} = \frac{120}{9} = \frac{40}{3}$$ 8. **Since $y$ cannot be both $\frac{70}{9}$ and $\frac{40}{3}$, the vectors are not equal unless the problem assumes equality component-wise only for the first and second components or there is a typo.** 9. **Assuming the first and third components must be equal, the only way is if the vectors are scalar multiples with the same scalar. Since they differ, no solution exists unless we consider only the first and second components.** 10. **Solve for $x$ using the second component and $y = \frac{70}{9}$ (from first component):** $$-\frac{5}{4}x = \frac{9}{2}y = 35$$ Multiply both sides by $-\frac{4}{5}$: $$x = 35 \times -\frac{4}{5} = -28$$ **Final answers:** $$x = -28, \quad y = \frac{70}{9}$$ Note: The third component does not match, indicating a possible inconsistency or typo in the problem.