On shifting the origin to (p, q), the coordinates of point (2, –1) changes to (5, 2). Find p and q.

Answer

To solve the problem, we need to find the values of \( p \) and \( q \) when the origin is shifted from the point \( (0, 0) \) to the point \( (p, q) \). The given coordinates of the point change from \( (2, -1) \) to \( (5, 2) \).

### Step-by-Step Solution:

1. **Understanding the Shift of Origin**:
When we shift the origin from \( (0, 0) \) to \( (p, q) \), the relationship between the old coordinates \( (x, y) \) and the new coordinates \( (X, Y) \) can be expressed as:
\[ x = X + p \]
\[ y = Y + q \]

2. **Substituting the Given Coordinates**:
We know the old coordinates \( (x, y) = (2, -1) \) and the new coordinates \( (X, Y) = (5, 2) \). We can substitute these values into the equations:
- For the x-coordinate:
\[ 2 = 5 + p \]
- For the y-coordinate:
\[ -1 = 2 + q \]

3. **Solving for \( p \)**:
Rearranging the first equation to solve for \( p \):
\[ p = 2 - 5 \]
\[ p = -3 \]

4. **Solving for \( q \)**:
Rearranging the second equation to solve for \( q \):
\[ q = -1 - 2 \]
\[ q = -3 \]

5. **Final Result**:
Therefore, the values of \( p \) and \( q \) are:
\[ (p, q) = (-3, -3) \]

### Conclusion:
The new origin after shifting is at the point \( (-3, -3) \).