1. **State the problem:** We need to find values of $x$, $y$, and $z$ such that the matrices $$\begin{bmatrix} x & 6 \\ -6 & 5 \\ -10 & 5 \end{bmatrix} = \begin{bmatrix} -7 & 6 \\ y-9 & 5 \\ -10 & z+x \end{bmatrix}$$ are equal. 2. **Recall the rule for matrix equality:** Two matrices are equal if and only if their corresponding elements are equal. 3. **Set up equations by equating corresponding elements:** - From position (1,1): $x = -7$ - From position (1,2): $6 = 6$ (already equal) - From position (2,1): $-6 = y - 9$ - From position (2,2): $5 = 5$ (already equal) - From position (3,1): $-10 = -10$ (already equal) - From position (3,2): $5 = z + x$ 4. **Solve for $y$:** $$-6 = y - 9 \implies y = -6 + 9 = 3$$ 5. **Solve for $z$:** We know $x = -7$, so $$5 = z + (-7) \implies z = 5 + 7 = 12$$ 6. **Final answers:** $$x = -7, \quad y = 3, \quad z = 12$$