Find the equation of the line which passes through the point `(5,6)` and the intercepts on the axes are equal in magnitude but opposite in sign.

Answer

To find the equation of the line that passes through the point (5, 6) and has intercepts on the axes that are equal in magnitude but opposite in sign, we can follow these steps:

### Step 1: Understand the condition of intercepts
Since the intercepts on the axes are equal in magnitude but opposite in sign, we can denote the x-intercept as \( a \) and the y-intercept as \( -a \). Thus, the intercepts can be represented as:
- x-intercept: \( a \)
- y-intercept: \( -a \)

### Step 2: Write the equation of the line in intercept form
The equation of a line in intercept form is given by:
\[ \frac{x}{a} + \frac{y}{-a} = 1 \]
This simplifies to:
\[ \frac{x}{a} - \frac{y}{a} = 1 \]
Multiplying through by \( a \) gives:
\[ x - y = a \]

### Step 3: Use the point (5, 6) to find \( a \)
Since the line passes through the point (5, 6), we can substitute \( x = 5 \) and \( y = 6 \) into the equation \( x - y = a \):
\[ 5 - 6 = a \]
This simplifies to:
\[ a = -1 \]

### Step 4: Substitute \( a \) back into the equation
Now that we have \( a = -1 \), we can substitute it back into the equation \( x - y = a \):
\[ x - y = -1 \]

### Step 5: Rearrange to standard form
Rearranging the equation gives us:
\[ x - y + 1 = 0 \]

### Final Answer
The equation of the line is:
\[ x - y + 1 = 0 \]

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