The equation of the plane with intercepts, 2,5 and 4 on the x,y and z axis respectively is .............

Answer

To find the equation of the plane with given intercepts on the x, y, and z axes, we can use the intercept form of the equation of a plane. The intercepts given are:

- x-intercept (a) = 2
- y-intercept (b) = 5
- z-intercept (c) = 4

The general form of the equation of a plane with intercepts on the axes is given by:

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

### Step 1: Substitute the intercepts into the equation

Substituting the values of the intercepts into the equation:

\[ \frac{x}{2} + \frac{y}{5} + \frac{z}{4} = 1 \]

### Step 2: Clear the denominators

To eliminate the fractions, we can multiply the entire equation by the least common multiple (LCM) of the denominators (2, 5, and 4). The LCM of 2, 5, and 4 is 20.

Multiplying through by 20 gives:

\[ 20 \cdot \left(\frac{x}{2}\right) + 20 \cdot \left(\frac{y}{5}\right) + 20 \cdot \left(\frac{z}{4}\right) = 20 \]

This simplifies to:

\[ 10x + 4y + 5z = 20 \]

### Step 3: Write the final equation

Thus, the equation of the plane with intercepts 2, 5, and 4 on the x, y, and z axes respectively is:

\[ 10x + 4y + 5z = 20 \]

### Final Answer:
The equation of the plane is \( 10x + 4y + 5z = 20 \).

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