Determine whether the following pair of lines is parallel, intersects at a single point, or is skew. If the lines are parallel, determine whether they are the same line (and thus intersect at all points). If the lines intersect at a single point, determine the point of intersection. \( x=2, y=7-t, z=1+t \) and \( x=-5-7 s, y=1+4 s, z=5-s \) Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice. A. The lines intersect at the single point (Simplify your answers.) B. The lines intersect at all points on the line \( r(t) \). C. The lines are parallel. D. The lines are skew.

To determine the relationship between the two lines: **First Line:** \[ x = 2, \quad y = 7 - t, \quad z = 1 + t \] This can be written in vector form as: \[ \mathbf{r}(t) = (2, 7, 1) + t(0, -1, 1) \] **Direction Vector:** \(\mathbf{d}_1 = (0, -1, 1)\) **Second Line:** \[ x = -5 - 7s, \quad y = 1 + 4s, \quad z = 5 - s \] This can be written in vector form as: \[ \mathbf{s}(s) = (-5, 1, 5) + s(-7, 4, -1) \] **Direction Vector:** \(\mathbf{d}_2 = (-7, 4, -1)\) **Step 1: Check for Parallelism** Two lines are parallel if their direction vectors are scalar multiples of each other. Check if there exists a scalar \(\lambda\) such that: \[ (0, -1, 1) = \lambda(-7, 4, -1) \] Equating components: \[ 0 = -7\lambda \quad \Rightarrow \quad \lambda = 0 \\ -1 = 4\lambda \quad \Rightarrow \quad \lambda = -\frac{1}{4} \\ 1 = -\lambda \quad \Rightarrow \quad \lambda = -1 \] The values of \(\lambda\) are inconsistent, so the lines are not parallel. **Step 2: Check for Intersection** Solve the system of equations to find if there's a common point: \[ \begin{cases} 2 = -5 - 7s \\ 7 - t = 1 + 4s \\ 1 + t = 5 - s \end{cases} \] From the first equation: \[ 2 = -5 - 7s \quad \Rightarrow \quad 7s = -7 \quad \Rightarrow \quad s = -1 \] Substitute \(s = -1\) into the second equation: \[ 7 - t = 1 + 4(-1) \quad \Rightarrow \quad 7 - t = -3 \quad \Rightarrow \quad t = 10 \] Substitute \(s = -1\) and \(t = 10\) into the third equation: \[ 1 + 10 = 5 - (-1) \quad \Rightarrow \quad 11 \neq 6 \] This inconsistency means there is no common point of intersection. **Conclusion:** Since the lines are neither parallel nor intersecting, they are **skew**. **Answer:** **D. The lines are skew.**