We are given that ABCD is a parallelogram, and we need to find the coordinates of point $D(x, y)$ such that $\overrightarrow{AD} = \overrightarrow{BC}$.

Step 1: Use the property of parallelograms

In a parallelogram, the vector $\overrightarrow{AD}$ is equal to the vector $\overrightarrow{BC}$. This gives us the equation:

$\overrightarrow{AD} = \overrightarrow{BC}$

Step 2: Find the vector $\overrightarrow{BC}$

The vector $\overrightarrow{BC}$ is calculated as the difference between the coordinates of point $C$ and point $B$:

$\overrightarrow{BC} = \left( C_x - B_x, C_y - B_y \right) = \left( 4 - 2, 3 - 1 \right) = (2, 2)$

Thus, $\overrightarrow{BC} = (2, 2)$.

Step 3: Use the vector $\overrightarrow{AD} = \overrightarrow{BC}$

Now, the vector $\overrightarrow{AD}$ is given by:

$\overrightarrow{AD} = \left( D_x - A_x, D_y - A_y \right)$

We are told that $\overrightarrow{AD} = \overrightarrow{BC} = (2, 2)$, so:

$(D_x - A_x, D_y - A_y) = (2, 2)$

Substitute the coordinates of point $A(-3, 1)$:

$(D_x - (-3), D_y - 1) = (2, 2)$

This simplifies to:

$(D_x + 3, D_y - 1) = (2, 2)$

Step 4: Solve for $D_x$ and $D_y$

Now, solve each component:

1. $D_x + 3 = 2 \Rightarrow D_x = 2 - 3 = -1$
2. $D_y - 1 = 2 \Rightarrow D_y = 2 + 1 = 3$

Step 5: Final coordinates of point $D$

Thus, the coordinates of point $D$ are $(-1, 3)$.

Final Answer:

The coordinates of point $D$ are $\boxed{(-1, 3)}$.

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Would you like further details or clarifications? Here are some related questions:

1. What is the general condition for the vectors of a parallelogram to be equal?
2. How do you find the area of a parallelogram given four points?
3. How do you prove that a quadrilateral is a parallelogram using vector properties?
4. How can you determine the midpoint of a parallelogram's diagonal?
5. How do you find the equation of the line passing through two points, given their coordinates?

Tip: Always remember that in a parallelogram, opposite sides are parallel and equal in length, which can help simplify vector-based problems.