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: Understand the Translation of Axes
When we shift the origin to a new point \((h, k)\), the new coordinates \((X, Y)\) are related to the old coordinates \((x, y)\) by the equations:
\[ x = X + h \]
\[ y = Y + k \]
In our case, \(h = 3\) and \(k = -1\).

### Step 2: Substitute the New Coordinates
Substituting \(h\) and \(k\) into the equations, we have:
\[ x = X + 3 \]
\[ y = Y - 1 \]

### Step 3: Substitute into the Original Equation
Now, we will substitute these expressions for \(x\) and \(y\) into the original equation \(2x - 3y + 5 = 0\):
\[ 2(X + 3) - 3(Y - 1) + 5 = 0 \]

### Step 4: Simplify the Equation
Now, we simplify the equation:
\[ 2(X + 3) - 3(Y - 1) + 5 = 0 \]
Expanding this, we get:
\[ 2X + 6 - 3Y + 3 + 5 = 0 \]
Combining like terms:
\[ 2X - 3Y + 14 = 0 \]

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

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