If origin (0, 0) shifted to (–3, 4), what will be the coordinates of the point in the new position of axes (i.e., new coordinates system) which is represented by (4, 2) in the old position of axes (i.e., old coordinate system)?

Answer

To find the new coordinates of the point (4, 2) after the origin has been shifted from (0, 0) to (-3, 4), we can follow these steps:

### Step 1: Identify the old coordinates and the new origin
- The old coordinates of the point are (4, 2).
- The new origin after the shift is (-3, 4).

### Step 2: Define the new coordinates
Let the new coordinates be represented as (a, b). We need to find the values of a and b.

### Step 3: Use the transformation formulas
The transformation from the old coordinates (x, y) to the new coordinates (a, b) can be expressed as:
- \( a = x - h \)
- \( b = y - k \)

Where:
- \( (h, k) \) is the new origin, which is (-3, 4).
- \( (x, y) \) are the old coordinates, which are (4, 2).

### Step 4: Substitute the values into the formulas
1. For \( a \):
\[ a = x - h = 4 - (-3) = 4 + 3 = 7 \]

2. For \( b \):
\[ b = y - k = 2 - 4 = -2 \]

### Step 5: Write the new coordinates
Thus, the new coordinates of the point in the new coordinate system are (7, -2).

### Final Answer
The coordinates of the point in the new position of axes are (7, -2).

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