The plane 2x+3y-4z=5 intersects the x-axis at (a,0,0), the y-axis at (0,b,0), and the z-axis at (0,0,c). The value of a+b+c is

Options

A. 1
B. `(35)/(12)`
C. 5
D. `(65)/(12)`

Explanation

To find the value of \( a + b + c \) where the plane \( 2x + 3y - 4z = 5 \) intersects the axes, we will determine the coordinates of the intersection points on the x-axis, y-axis, and z-axis.

### Step-by-step Solution:

1. **Find the intersection with the x-axis:**
- On the x-axis, \( y = 0 \) and \( z = 0 \).
- Substitute \( y = 0 \) and \( z = 0 \) into the plane equation:
\[ 2x + 3(0) - 4(0) = 5 \implies 2x = 5 \implies x = \frac{5}{2} \]
- Therefore, the intersection point on the x-axis is \( (a, 0, 0) \) where \( a = \frac{5}{2} \).

2. **Find the intersection with the y-axis:**
- On the y-axis, \( x = 0 \) and \( z = 0 \).
- Substitute \( x = 0 \) and \( z = 0 \) into the plane equation:
\[ 2(0) + 3y - 4(0) = 5 \implies 3y = 5 \implies y = \frac{5}{3} \]
- Therefore, the intersection point on the y-axis is \( (0, b, 0) \) where \( b = \frac{5}{3} \).

3. **Find the intersection with the z-axis:**
- On the z-axis, \( x = 0 \) and \( y = 0 \).
- Substitute \( x = 0 \) and \( y = 0 \) into the plane equation:
\[ 2(0) + 3(0) - 4z = 5 \implies -4z = 5 \implies z = -\frac{5}{4} \]
- Therefore, the intersection point on the z-axis is \( (0, 0, c) \) where \( c = -\frac{5}{4} \).

4. **Calculate \( a + b + c \):**
- Now we can add the values of \( a \), \( b \), and \( c \):
\[ a + b + c = \frac{5}{2} + \frac{5}{3} - \frac{5}{4} \]

5. **Find a common denominator:**
- The least common multiple of 2, 3, and 4 is 12.
- Convert each fraction:
\[ a = \frac{5}{2} = \frac{5 \times 6}{2 \times 6} = \frac{30}{12} \]
\[ b = \frac{5}{3} = \frac{5 \times 4}{3 \times 4} = \frac{20}{12} \]
\[ c = -\frac{5}{4} = -\frac{5 \times 3}{4 \times 3} = -\frac{15}{12} \]

6. **Add the fractions:**
\[ a + b + c = \frac{30}{12} + \frac{20}{12} - \frac{15}{12} = \frac{30 + 20 - 15}{12} = \frac{35}{12} \]

### Final Answer:
Thus, the value of \( a + b + c \) is \( \frac{35}{12} \).