Find the equation of the line, which passes through (2, –5) and cuts off equal intercepts on both the axes.

Answer

To find the equation of the line that passes through the point (2, -5) and cuts off equal intercepts on both axes, we can follow these steps:

### Step 1: Understand the form of the equation
Since the line cuts off equal intercepts on both axes, we can express the equation of the line in the intercept form:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
where \( a \) and \( b \) are the x-intercept and y-intercept respectively. Given that the intercepts are equal, we can set \( a = b \).

### Step 2: Substitute equal intercepts
Let \( a = b = c \). Therefore, the equation can be rewritten as:
\[ \frac{x}{c} + \frac{y}{c} = 1 \]
This simplifies to:
\[ x + y = c \]

### Step 3: Use the point (2, -5)
Since the line passes through the point (2, -5), we can substitute these coordinates into the equation:
\[ 2 + (-5) = c \]
This simplifies to:
\[ 2 - 5 = c \implies c = -3 \]

### Step 4: Write the equation of the line
Now that we have \( c = -3 \), we can write the equation of the line:
\[ x + y = -3 \]

### Step 5: Rearranging the equation
We can rearrange this equation into the standard form:
\[ x + y + 3 = 0 \]

### Final Answer
The equation of the line that passes through (2, -5) and cuts off equal intercepts on both axes is:
\[ x + y + 3 = 0 \]

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