Find the equation of a line which passes through `(-3,2)` and makes intercepts equal in magnitude but opposite in sign on `X` and `Y`-axis.

Answer

To find the equation of a line that passes through the point \((-3, 2)\) and makes intercepts equal in magnitude but opposite in sign on the X and Y axes, we can follow these steps:

### Step 1: Understand the intercepts
The line makes intercepts equal in magnitude but opposite in sign on the X and Y axes. This means if the X-intercept is \(-a\), then the Y-intercept will be \(a\).

### Step 2: Write the equation of the line
The general form of the equation of a line with X-intercept \(a\) and Y-intercept \(b\) is given by:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
In our case, since the intercepts are \(-a\) and \(a\), we can substitute \(b = a\) and \(a = -a\):
\[ \frac{x}{-a} + \frac{y}{a} = 1 \]
This simplifies to:
\[ -\frac{x}{a} + \frac{y}{a} = 1 \]
Multiplying through by \(a\) (assuming \(a \neq 0\)):
\[ -y + x = a \]
Rearranging gives:
\[ x - y = a \quad \text{(Equation 1)} \]

### Step 3: Substitute the point into the equation
Since the line passes through the point \((-3, 2)\), we can substitute \(x = -3\) and \(y = 2\) into Equation 1:
\[ -3 - 2 = a \]
This simplifies to:
\[ -5 = a \]

### Step 4: Substitute \(a\) back into the line equation
Now that we have \(a = -5\), we can substitute this back into the equation \(x - y = a\):
\[ x - y = -5 \]
This can be rearranged to:
\[ y = x + 5 \]

### Final Answer
Thus, the equation of the line is:
\[ y = x + 5 \]