To find the equation of a line that passes through two given points, we can use the point-slope form of the line equation, $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept.

The slope $m$ is calculated as: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Let's go through each pair of points:

(a) Points: $(0, 0)$ and $(3, -12)$

1. Calculate the slope $m$: $m = \frac{-12 - 0}{3 - 0} = \frac{-12}{3} = -4$

2. Equation of the line:
   Using the point $(0, 0)$, the equation is: $y = -4x$

(b) Points: $(-4, 2)$ and $(0, 0)$

1. Calculate the slope $m$: $m = \frac{0 - 2}{0 - (-4)} = \frac{-2}{4} = -\frac{1}{2}$

2. Equation of the line:
   Using the point $(0, 0)$, the equation is: $y = -\frac{1}{2}x$

(c) Points: $(1, 1)$ and $(4, 4)$

1. Calculate the slope $m$: $m = \frac{4 - 1}{4 - 1} = \frac{3}{3} = 1$

2. Equation of the line:
   Using the point $(1, 1)$, the equation is: $y = x$

(d) Points: $(-5, 3)$ and $(1, -9)$

1. Calculate the slope $m$: $m = \frac{-9 - 3}{1 - (-5)} = \frac{-12}{6} = -2$

2. Equation of the line:
   Using the point $(-5, 3)$, the equation is: $y = -2x - 7$

Summary of Equations

• (a) $y = -4x$
• (b) $y = -\frac{1}{2}x$
• (c) $y = x$
• (d) $y = -2x - 7$

Would you like further details on any of these calculations?

Related Questions

1. How do you find the slope when given two points?
2. What is the point-slope form of a line equation?
3. How do you find the y-intercept if you know the slope and one point?
4. How would the equation change if the points were reversed?
5. How can you verify that a point lies on a specific line?

Tip

For any two points, you can always calculate the slope to determine if the line is increasing, decreasing, or horizontal.