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.