The intercepts of the plane `2x-3y+4z=12` on the coordinate axes are given by

Options

A. `3,-2,1.5`
B. `6,-4,3`
C. `6,-4,-3`
D. `2,-3,4`

Explanation

To find the intercepts of the plane given by the equation \(2x - 3y + 4z = 12\) on the coordinate axes, we can follow these steps:

### Step 1: Write the equation in intercept form
The intercept form of a plane is given by:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]
where \(a\), \(b\), and \(c\) are the x-intercept, y-intercept, and z-intercept, respectively.

### Step 2: Divide the entire equation by 12
To convert the given equation into intercept form, we need to express it such that the right-hand side equals 1. We start with the original equation:
\[ 2x - 3y + 4z = 12 \]
Now, divide every term by 12:
\[ \frac{2x}{12} - \frac{3y}{12} + \frac{4z}{12} = 1 \]
This simplifies to:
\[ \frac{x}{6} - \frac{y}{4} + \frac{z}{3} = 1 \]

### Step 3: Rearrange the equation
Rearranging gives us:
\[ \frac{x}{6} + \frac{y}{-4} + \frac{z}{3} = 1 \]

### Step 4: Identify the intercepts
From the equation \(\frac{x}{6} + \frac{y}{-4} + \frac{z}{3} = 1\), we can identify:
- \(a = 6\) (x-intercept)
- \(b = -4\) (y-intercept)
- \(c = 3\) (z-intercept)

### Step 5: Write the final intercepts
Thus, the intercepts on the coordinate axes are:
- **x-intercept**: \(6\)
- **y-intercept**: \(-4\)
- **z-intercept**: \(3\)

### Final Answer:
The intercepts of the plane \(2x - 3y + 4z = 12\) on the coordinate axes are:
- x-intercept: \(6\)
- y-intercept: \(-4\)
- z-intercept: \(3\)

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