If plane `ax+y+z=7` has equal intercepts on axes, then `a` is equal to

Options

A. 7
B. `(1)/(7)`
C. 1
D. `(1)/(5)`

Answer

To solve the problem, we need to find the value of \( a \) such that the plane given by the equation \( ax + y + z = 7 \) has equal intercepts on the axes.

### Step-by-Step Solution:

1. **Understanding the Plane Equation**:
The equation of the plane is given as:
\[ ax + y + z = 7 \]
We want to express this in a form that reveals the intercepts.

2. **Rearranging the Equation**:
We can rearrange the equation to express it in the standard intercept form:
\[ \frac{x}{\frac{7}{a}} + \frac{y}{7} + \frac{z}{7} = 1 \]
Here, the intercepts on the axes can be identified as:
- \( A = \frac{7}{a} \) (x-intercept)
- \( B = 7 \) (y-intercept)
- \( C = 7 \) (z-intercept)

3. **Setting the Condition for Equal Intercepts**:
For the plane to have equal intercepts, we need:
\[ A = B = C \]
Substituting the values of the intercepts:
\[ \frac{7}{a} = 7 \]

4. **Solving for \( a \)**:
To find \( a \), we can set up the equation:
\[ \frac{7}{a} = 7 \]
Cross-multiplying gives:
\[ 7 = 7a \]
Dividing both sides by 7:
\[ 1 = a \]

5. **Conclusion**:
Therefore, the value of \( a \) is:
\[ a = 1 \]

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