If a plane through the points `(2,0,0) , (0,3,0) AND (0,0,4)` the equation of plane is `"_______."`

Answer

To find the equation of the plane passing through the points \( (2,0,0) \), \( (0,3,0) \), and \( (0,0,4) \), we can use the general form of the equation of a plane that intersects the coordinate axes at points \( (a,0,0) \), \( (0,b,0) \), and \( (0,0,c) \). The equation of such a plane is given by:

\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]

### Step-by-Step Solution:

1. **Identify the intercepts**:
- The plane passes through the points \( (2,0,0) \), \( (0,3,0) \), and \( (0,0,4) \).
- Here, \( a = 2 \), \( b = 3 \), and \( c = 4 \).

2. **Substitute the values into the equation**:
- Substitute \( a \), \( b \), and \( c \) into the plane equation:
\[ \frac{x}{2} + \frac{y}{3} + \frac{z}{4} = 1 \]

3. **Multiply through by the least common multiple (LCM)**:
- The LCM of the denominators \( 2, 3, \) and \( 4 \) is \( 12 \).
- Multiply the entire equation by \( 12 \) to eliminate the fractions:
\[ 12 \left( \frac{x}{2} \right) + 12 \left( \frac{y}{3} \right) + 12 \left( \frac{z}{4} \right) = 12 \]
- This simplifies to:
\[ 6x + 4y + 3z = 12 \]

4. **Final equation of the plane**:
- Rearranging gives us the final equation of the plane:
\[ 6x + 4y + 3z - 12 = 0 \]

### Final Answer:
The equation of the plane is:
\[ 6x + 4y + 3z - 12 = 0 \]