To find the equation of a line that passes through two given points $(1, -5)$ and $(4, 4)$, we can use the point-slope form of a line equation.

Step 1: Find the slope

The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated by the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the given points $(x_1, y_1) = (1, -5)$ and $(x_2, y_2) = (4, 4)$:

$m = \frac{4 - (-5)}{4 - 1} = \frac{4 + 5}{4 - 1} = \frac{9}{3} = 3$

So, the slope $m$ is 3.

Step 2: Use the point-slope form

The point-slope form of a line is given by:

$y - y_1 = m(x - x_1)$

Using the slope $m = 3$ and the point $(1, -5)$, we substitute into the point-slope equation:

$y - (-5) = 3(x - 1)$

Simplifying:

$y + 5 = 3(x - 1)$

Step 3: Simplify to slope-intercept form

Now, distribute and simplify:

$y + 5 = 3x - 3$

Subtract 5 from both sides:

$y = 3x - 3 - 5$

$y = 3x - 8$

Final Answer:

The equation of the line is:

$y = 3x - 8$

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. What is the equation of a line passing through the points $(2, 3)$ and $(5, -6)$?
2. How do you find the slope of a line parallel to $y = 2x + 3$?
3. What is the equation of a line with slope $4$ passing through the point $(2, 1)$?
4. How do you determine if two lines are perpendicular using their slopes?
5. What is the slope-intercept form of a vertical line passing through $x = 2$?

Tip: Remember that for vertical lines, the slope is undefined, and the equation is always of the form $x = c$.