1. **Problem Statement:** Given the vector equation $(x - y, x + y, z - 1) = (4, 2, 3)$, find the values of $x$, $y$, and $z$. 2. **Understanding Vector Equality:** Two vectors are equal if and only if their corresponding components are equal. This means: $$x - y = 4$$ $$x + y = 2$$ $$z - 1 = 3$$ 3. **Solving for $x$ and $y$:** Add the first two equations to eliminate $y$: $$ (x - y) + (x + y) = 4 + 2 $$ $$ 2x = 6 $$ $$ x = 3 $$ Substitute $x = 3$ into the second equation: $$ 3 + y = 2 $$ $$ y = 2 - 3 = -1 $$ 4. **Solving for $z$:** From the third equation: $$ z - 1 = 3 $$ $$ z = 3 + 1 = 4 $$ 5. **Final Answer:** $$ x = 3, \quad y = -1, \quad z = 4 $$ These values satisfy the given vector equality.