Find the equation of the plane passing through the point (-2,1,-3) and making equal intercept on the coordinate axes

Answer

To find the equation of the plane that passes through the point (-2, 1, -3) and makes equal intercepts on the coordinate axes, we can follow these steps:

### Step 1: Understand the intercept form of the plane equation
The equation of a plane in intercept form is given by:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]
where \(a\), \(b\), and \(c\) are the intercepts on the x, y, and z axes respectively.

### Step 2: Set equal intercepts
Since the plane makes equal intercepts on the coordinate axes, we can set \(a = b = c = k\). Therefore, the equation becomes:
\[ \frac{x}{k} + \frac{y}{k} + \frac{z}{k} = 1 \]
This simplifies to:
\[ \frac{x + y + z}{k} = 1 \]
Multiplying through by \(k\) gives:
\[ x + y + z = k \]

### Step 3: Substitute the point into the equation
Now, we need to find the value of \(k\) such that the plane passes through the point (-2, 1, -3). We substitute \(x = -2\), \(y = 1\), and \(z = -3\) into the equation:
\[ -2 + 1 - 3 = k \]
Calculating the left side:
\[ -2 + 1 - 3 = -4 \]
Thus, we have:
\[ k = -4 \]

### Step 4: Write the final equation of the plane
Substituting \(k\) back into the equation of the plane gives:
\[ x + y + z = -4 \]

### Final Answer
The equation of the plane is:
\[ x + y + z = -4 \]