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To determine which points lie on the line \( y = 2x - 5 \), we will substitute the coordinates of each point into the equation and check if the equation holds true.

### Given equation:
\[ y = 2x - 5 \]

### Points to check:
1. \( (3, 1) \)
2. \( (-4, -12) \)
3. \( (-2, -9) \)
4. \( (7, 9) \)

Let's check each point:

1. **For \( (3, 1) \):**
   - Substitute \( x = 3 \) into the equation:
   \[ y = 2(3) - 5 = 6 - 5 = 1 \]
   - The y-value is 1, which matches the given point \( (3, 1) \). So, this point lies on the line.

2. **For \( (-4, -12) \):**
   - Substitute \( x = -4 \) into the equation:
   \[ y = 2(-4) - 5 = -8 - 5 = -13 \]
   - The y-value is -13, but the point given is \( (-4, -12) \). So, this point does not lie on the line.

3. **For \( (-2, -9) \):**
   - Substitute \( x = -2 \) into the equation:
   \[ y = 2(-2) - 5 = -4 - 5 = -9 \]
   - The y-value is -9, which matches the given point \( (-2, -9) \). So, this point lies on the line.

4. **For \( (7, 9) \):**
   - Substitute \( x = 7 \) into the equation:
   \[ y = 2(7) - 5 = 14 - 5 = 9 \]
   - The y-value is 9, which matches the given point \( (7, 9) \). So, this point lies on the line.

### Conclusion:
The points that lie on the line \( y = 2x - 5 \) are:
- \( (3, 1) \)
- \( (-2, -9) \)
- \( (7, 9) \)

Would you like more details or have any other questions?

### Follow-up questions:
1. What is the slope of the line \( y = 2x - 5 \) and how is it determined?
2. How would the equation change if the line were parallel but passed through a different point?
3. What would be the equation of the line perpendicular to \( y = 2x - 5 \)?
4. If the line intersects the x-axis, what is the x-intercept?
5. How would you find the distance between two points on this line?

### Tip:
Always verify if a point lies on a line by substituting the coordinates into the line's equation.

Choose all the points which lie on the line y = 2x - 5: (3,1), (-4,-12), (-2,-9), (7,9).

Learn how to determine which points lie on the line y = 2x - 5 by substituting the x-values of each point into the equation. This exercise involves basic algebra and understanding the point-slope form of a linear equation, suitable for Grades 7-10.

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