Let $S$ be a region in the plane with area 10.  When we apply the matrix
\[\begin{pmatrix} 2 & 1 \\ 7 & -3 \end{pmatrix}\]to $S,$ we obtain the region $S'.$  Find the area of $S'.$
Note that
\[\begin{vmatrix} 2 & 1 \\ 7 & -3 \end{vmatrix} = (2)(-3) - (1)(7) = -13,\]so the matrix scales the area of any region by a factor of $|-13| = 13.$  In particular, the area of $S'$ is $13 \cdot 10 = \boxed{130}.$
130
Level 3
Precalculus
Problem: Let $S$ be a region in the plane with area 10.  When we apply the matrix
\[\begin{pmatrix} 2 & 1 \\ 7 & -3 \end{pmatrix}\]to $S,$ we obtain the region $S'.$  Find the area of $S'.$
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the area of the transformed region $S'$ after applying the matrix, we need to understand how the area changes under linear transformations represented by matrices. The area of a region transformed by a matrix is scaled by the absolute value of the determinant of that matrix.

First, we calculate the determinant of the matrix 
\[\begin{pmatrix} 2 & 1 \\ 7 & -3 \end{pmatrix}.\]
The determinant is computed as follows:
\[
\text{det} = (2)(-3) - (1)(7) = -6 - 7 = -13.
\]
Next, we take the absolute value of the determinant:
\[
|\text{det}| = |-13| = 13.
\]
This means that the area of the region $S'$ is 13 times the area of the original region $S$.

Given that the area of region $S$ is 10, we can find the area of region $S'$ by multiplying the area of $S$ by the absolute value of the determinant:
\[
\text{Area of } S' = |\text{det}| \times \text{Area of } S = 13 \times 10 = 130.
\]
Thus, the final answer is 130.<|END_OF_LATENT|> The final answer is $130$.