What is the equation of a plane through the x-axis and passing through the point `(1,2,3)`?

Options

A. `x+y+z=6`
B. `x=1`
C. `y+z=5`
D. `z+y=1`

Explanation

To find the equation of a plane that passes through the x-axis and the point (1, 2, 3), we can follow these steps:

### Step 1: Understand the properties of the plane
A plane that passes through the x-axis means that for any point on the x-axis, the plane will include that point. The x-axis can be represented by points of the form (x, 0, 0) where x can be any real number.

### Step 2: General form of the equation of a plane
The general equation of a plane can be expressed as:
\[ Ax + By + Cz + D = 0 \]
where A, B, C, and D are constants.

### Step 3: Since the plane passes through the x-axis
Since the plane passes through the x-axis, any point (x, 0, 0) must satisfy the plane's equation. This means that when we substitute y = 0 and z = 0 into the plane's equation, it should hold true for all x. This leads us to conclude that the coefficient of x (A) must be 0, otherwise, we would get a contradiction for different values of x.

### Step 4: Simplifying the equation
Thus, the equation simplifies to:
\[ By + Cz + D = 0 \]
This means that the plane can be expressed in terms of y and z.

### Step 5: Use the point (1, 2, 3)
Now we need to ensure that the plane also passes through the point (1, 2, 3). Substituting x = 1, y = 2, and z = 3 into the equation gives us:
\[ B(2) + C(3) + D = 0 \]
This equation must hold true for the plane.

### Step 6: Choosing values for B and C
To find a specific equation, we can choose values for B and C. For simplicity, let's choose B = 1 and C = -1. Then we have:
\[ 2(1) + 3(-1) + D = 0 \]
This simplifies to:
\[ 2 - 3 + D = 0 \]
\[ D = 1 \]

### Step 7: Final equation of the plane
Substituting B, C, and D back into the plane equation gives us:
\[ y - z + 1 = 0 \]
or
\[ y - z = -1 \]

Thus, the equation of the plane is:
\[ y - z = -1 \]