1. **State the problem:** We have the matrix equation $$\begin{bmatrix}9 & 3x+1 \\ 2y-1 & 10\end{bmatrix} = \begin{bmatrix}9 & 16 \\ -5 & 10\end{bmatrix}$$ and need to find values of $x$ and $y$ that make this true. 2. **Recall matrix equality rule:** Two matrices are equal if and only if their corresponding elements are equal. 3. **Set corresponding elements equal:** - Top left: $9 = 9$ (already true) - Top right: $3x + 1 = 16$ - Bottom left: $2y - 1 = -5$ - Bottom right: $10 = 10$ (already true) 4. **Solve for $x$:** $$3x + 1 = 16$$ Subtract 1 from both sides: $$3x + \cancel{1} - \cancel{1} = 16 - 1$$ $$3x = 15$$ Divide both sides by 3: $$\frac{3x}{\cancel{3}} = \frac{15}{\cancel{3}}$$ $$x = 5$$ 5. **Solve for $y$:** $$2y - 1 = -5$$ Add 1 to both sides: $$2y - \cancel{1} + \cancel{1} = -5 + 1$$ $$2y = -4$$ Divide both sides by 2: $$\frac{2y}{\cancel{2}} = \frac{-4}{\cancel{2}}$$ $$y = -2$$ 6. **Final answer:** The values that satisfy the matrix equation are $x = 5$ and $y = -2$.