Find an equation of the line where the planes $$ Q $$ and $$ R $$ intersect. Q: $$ -2 x-3 y+4 z=1 $$; $$ R: x+y+z=1 $$ Choose the correct answer below. A. An equation of the line is given by $$ x=-\frac{1}{3}-3 t, y=-2 t, z=\frac{1}{4}+4 t $$, where $$ -\infty

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To find the equation of the line where the planes $$ Q $$ and $$ R $$ intersect, we'll follow these steps:

1. **Identify the Equations of the Planes:**
   - Plane $$ Q $$: $$ -2x - 3y + 4z = 1 $$
   - Plane $$ R $$: $$ x + y + z = 1 $$

2. **Find the Direction Vector of the Line:**
   The direction vector $$ \mathbf{v} $$ of the intersection line is perpendicular to the normals of both planes. The normal vectors are:
   - $$ \mathbf{n}_Q = (-2, -3, 4) $$
   - $$ \mathbf{n}_R = (1, 1, 1) $$

The direction vector $$ \mathbf{v} $$ is given by the cross product $$ \mathbf{n}_Q \times \mathbf{n}_R $$:
   $$
   \mathbf{v} = \begin{vmatrix}
   \mathbf{i} & \mathbf{j} & \mathbf{k} \
   -2 & -3 & 4 \
   1 & 1 & 1 \
   \end{vmatrix} = (-7, 6, 1)
   $$

3. **Find a Point on the Line:**
   Solve the system of equations to find a specific point on both planes. Let's set $$ z = 0 $$:
   $$
   \begin{cases}
   -2x - 3y = 1 \
   x + y = 1 \
   \end{cases}
   $$

Solving these equations:
   - From the second equation: $$ y = 1 - x $$
   - Substitute into the first equation:
     $$
     -2x - 3(1 - x) = 1 \implies -2x - 3 + 3x = 1 \implies x = 4
     $$
   - Then $$ y = 1 - 4 = -3 $$

So, one point on the line is $$ (4, -3, 0) $$.

4. **Write the Parametric Equations of the Line:**
   Using the direction vector and the point:
   $$
   \begin{cases}
   x = 4 - 7t \
   y = -3 + 6t \
   z = t \
   \end{cases} \quad \text{where} \quad t \in \mathbb{R}
   $$

**Therefore, the correct answer is:**

**C. An equation of the line is given by $$ x = 4 - 7 t $$, $$ y = -3 + 6 t $$, $$ z = t $$, where $$ -\infty < t < \infty $$.**
The equation of the line where planes $$ Q $$ and $$ R $$ intersect is: $$ x = 4 - 7t, \quad y = -3 + 6t, \quad z = t $$ where $$ t $$ is any real number.

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