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HALLAR LAS COORDENADAS DEL PUNTO DE LA RECTA Y GRAFICA 

First, let's find the coordinates of the point on the line given by the equation:

\[
\frac{x-2}{3} = \frac{y+4}{-2} = \frac{z-1}{2}
\]

for \( x = 3 \).

1. **Substitute \( x = 3 \) into the equation:**

\[
\frac{3-2}{3} = \frac{y+4}{-2} = \frac{z-1}{2}
\]

This simplifies to:

\[
\frac{1}{3} = \frac{y+4}{-2} = \frac{z-1}{2}
\]

2. **Solve for \( y \):**

\[
\frac{1}{3} = \frac{y+4}{-2}
\]

Multiply both sides by \(-2\):

\[
-\frac{2}{3} = y + 4
\]

Subtract 4 from both sides:

\[
y = -\frac{2}{3} - 4 = -\frac{2}{3} - \frac{12}{3} = -\frac{14}{3}
\]

3. **Solve for \( z \):**

\[
\frac{1}{3} = \frac{z-1}{2}
\]

Multiply both sides by 2:

\[
\frac{2}{3} = z - 1
\]

Add 1 to both sides:

\[
z = \frac{2}{3} + 1 = \frac{2}{3} + \frac{3}{3} = \frac{5}{3}
\]

Therefore, the coordinates of the point on the line for \( x = 3 \) are:

\[
(3, -\frac{14}{3}, \frac{5}{3})
\]

Next, we will create a graph of the line along with the identified point.

Here is the graph showing the line and the point \((3, -\frac{14}{3}, \frac{5}{3})\):

Let's generate the graph for the line and the point.

I'll start by plotting the line in a 3D space and highlighting the specified point.

```python
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import numpy as np

# Parameters for the line
t = np.linspace(-10, 10, 400)
x = 3 * t + 2
y = -2 * t - 4
z = 2 * t + 1

# Point coordinates
x_point = 3
y_point = -14/3
z_point = 5/3

# Plotting the line and point
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot(x, y, z, label='Line')
ax.scatter(x_point, y_point, z_point, color='r', label='Point (3, -14/3, 5/3)')

ax.set_xlabel('X')
ax.set_ylabel('Y')
ax.set_zlabel('Z')
ax.legend()

plt.show()
```

The plot code will generate a 3D graph. 

Would you like more details or have any questions?

Here are 5 numbered questions you might find helpful:

1. How do you find the parametric equations of a line in 3D?
2. What is the significance of symmetric equations in representing lines in 3D?
3. How can you determine the intersection of two lines in 3D space?
4. What methods can you use to plot 3D graphs in Python?
5. How can you convert symmetric equations to parametric form?

**Tip:** When plotting in 3D, always ensure you label the axes clearly to interpret the graph correctly.

Learn how to find the coordinates of a point on a 3D line using parametric equations and graph plotting. This problem involves applying parametric equations in 3D geometry to determine specific coordinates. Suitable for advanced high school students studying 3D geometry and parametric equations.

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