If the plane `x + ay + z=5` has equal intercepts on axes, then the value of a is

Answer

To solve the problem, we need to find the value of \( a \) such that the plane \( x + ay + z = 5 \) has equal intercepts on the axes.

### Step-by-Step Solution:

1. **Understanding the Plane Equation**:
The given equation of the plane is:
\[ x + ay + z = 5 \]

2. **Convert to Intercept Form**:
To find the intercepts, we can rewrite the equation in the standard intercept form:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]
To do this, we divide the entire equation by 5:
\[ \frac{x}{5} + \frac{ay}{5} + \frac{z}{5} = 1 \]

3. **Identifying Intercepts**:
In the equation \( \frac{x}{5} + \frac{y}{\frac{5}{a}} + \frac{z}{5} = 1 \), we can identify the intercepts:
- The x-intercept is \( 5 \)
- The y-intercept is \( \frac{5}{a} \)
- The z-intercept is \( 5 \)

4. **Setting Equal Intercepts**:
Since the problem states that the intercepts are equal, we set the y-intercept equal to the x-intercept (and z-intercept):
\[ 5 = \frac{5}{a} \]

5. **Solving for \( a \)**:
To solve for \( a \), we can multiply both sides by \( a \):
\[ 5a = 5 \]
Now, divide both sides by 5:
\[ a = 1 \]

### Final Answer:
The value of \( a \) is \( 1 \).