Find the transformed equation of the straight line `2x - 3y+ 5= 0`, when the origin is shifted to the point `(3, -1)` after translation of axes.

Explanation

To find the transformed equation of the straight line \(2x - 3y + 5 = 0\) when the origin is shifted to the point \((3, -1)\), we will follow these steps:

### Step 1: Define the Old and New Coordinate Systems
Let the old coordinates be \((x, y)\) and the new coordinates be \((X, Y)\). The new origin is at the point \((3, -1)\).

### Step 2: Establish the Relationship Between Old and New Coordinates
When we shift the origin, the relationship between the old and new coordinates can be expressed as:
\[ x = X + 3 \]
\[ y = Y - 1 \]

### Step 3: Substitute the New Coordinates into the Original Equation
We will substitute \(x\) and \(y\) in the original equation \(2x - 3y + 5 = 0\) using the relationships established in Step 2:
\[ 2(X + 3) - 3(Y - 1) + 5 = 0 \]

### Step 4: Simplify the Equation
Now, we will simplify the equation step by step:
1. Expand the equation:
\[ 2X + 6 - 3Y + 3 + 5 = 0 \]
2. Combine like terms:
\[ 2X - 3Y + (6 + 3 + 5) = 0 \]
\[ 2X - 3Y + 14 = 0 \]

### Step 5: Write the Transformed Equation
Thus, the transformed equation of the straight line in the new coordinate system is:
\[ 2X - 3Y + 14 = 0 \]

### Summary
The transformed equation of the straight line \(2x - 3y + 5 = 0\) after shifting the origin to the point \((3, -1)\) is:
\[ 2X - 3Y + 14 = 0 \]