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To find the point of intersection of the two lines \( L_1 \) and \( L_2 \), we solve the parametric equations for \( t \) and \( s \) such that the coordinates match:

### Equations of the lines:
1. \( L_1: x = -2t, \, y = 1 + 2t, \, z = 3t \)
2. \( L_2: x = -6 + 2s, \, y = 5s, \, z = 1 + 5s \)

### Step-by-step solution:

#### 1. Set the \( x \)-coordinates equal:
\[
-2t = -6 + 2s
\]
Rearrange:
\[
2t + 2s = 6 \quad \implies \quad t + s = 3 \tag{1}
\]

#### 2. Set the \( y \)-coordinates equal:
\[
1 + 2t = 5s
\]
Rearrange:
\[
2t - 5s = -1 \tag{2}
\]

#### 3. Solve the system of equations \((1)\) and \((2)\):
From \((1)\): \( t = 3 - s \).

Substitute \( t = 3 - s \) into \((2)\):
\[
2(3 - s) - 5s = -1
\]
\[
6 - 2s - 5s = -1
\]
\[
6 - 7s = -1 \quad \implies \quad 7s = 7 \quad \implies \quad s = 1
\]

Substitute \( s = 1 \) into \((1)\):
\[
t + 1 = 3 \quad \implies \quad t = 2
\]

#### 4. Compute the intersection point:
Substitute \( t = 2 \) into \( L_1 \):
\[
x = -2t = -2(2) = -4, \quad y = 1 + 2t = 1 + 2(2) = 5, \quad z = 3t = 3(2) = 6
\]

Thus, the point of intersection is:
\[
P = (-4, 5, 6)
\]

Would you like further clarification or additional details?

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### Expanding Questions:
1. What happens if the lines do not intersect but are skew? How is this determined mathematically?
2. How do you find the angle between two intersecting lines in three dimensions?
3. Can parametric equations represent curves other than straight lines? Provide an example.
4. How can this approach be extended to solve for the intersection of a line and a plane?
5. How do you determine if two lines are parallel in three-dimensional space?

### Tip:
When solving for intersections of parametric lines, always confirm the solution satisfies all equations for consistency.

Find the point of intersection of the two lines L1: x = -2t, y = 1 + 2t, z = 3t and L2: x = -6 + 2s, y = 5s, z = 1 + 5s.

Learn how to find the intersection of two parametric lines with equations L1: x = -2t, y = 1 + 2t, z = 3t and L2: x = -6 + 2s, y = 5s, z = 1 + 5s. This solution involves solving a system of linear equations to determine the point of intersection in 3D space, suitable for students in Grades 10-12.

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