The origin is shifted to (2,3) by the translation of axes. If a point P has changed as
(4,-3), find the coordinates of P in the original system.

Answer

To find the coordinates of point P in the original coordinate system after the origin has been shifted to (2, 3), we can follow these steps:

### Step 1: Understand the Shift
The origin has been shifted from (0, 0) to (2, 3). This means that the new coordinates (capital X, capital Y) are related to the original coordinates (x, y) by the equations:
- \( X = x + \Delta x \)
- \( Y = y + \Delta y \)

Where \( \Delta x = 2 \) and \( \Delta y = 3 \).

### Step 2: Write Down the Given Information
We are given the new coordinates of point P in the shifted system:
- \( X = 4 \)
- \( Y = -3 \)

### Step 3: Set Up the Equations
Using the relationships from Step 1, we can set up the equations:
1. \( 4 = x + 2 \) (for the x-coordinate)
2. \( -3 = y + 3 \) (for the y-coordinate)

### Step 4: Solve for the Original x-coordinate
From the first equation:
\[ 4 = x + 2 \]
Subtract 2 from both sides:
\[ x = 4 - 2 \]
\[ x = 2 \]

### Step 5: Solve for the Original y-coordinate
From the second equation:
\[ -3 = y + 3 \]
Subtract 3 from both sides:
\[ y = -3 - 3 \]
\[ y = -6 \]

### Step 6: Write the Original Coordinates
Thus, the coordinates of point P in the original system (before the shift) are:
\[ (x, y) = (2, -6) \]

### Final Answer
The coordinates of P in the original system are \( (2, -6) \).

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