To find the equation of a line in slope-intercept form ($y = mx + b$), we need two pieces of information:

1. The slope ($m$)
2. The y-intercept ($b$)

Step 1: Find the slope $m$

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

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

Here, the two points are: $(2, -5) \quad \text{and} \quad (0, -2)$

Substitute the values into the slope formula: $m = \frac{-2 - (-5)}{0 - 2} = \frac{-2 + 5}{-2} = \frac{3}{-2} = -\frac{3}{2}$

Step 2: Use the slope-intercept form $y = mx + b$

Now that we have the slope, $m = -\frac{3}{2}$, we can substitute it into the slope-intercept form: $y = -\frac{3}{2}x + b$

Step 3: Find the y-intercept $b$

To find $b$, we can use one of the given points. Let's use the point $(0, -2)$, where $x = 0$ and $y = -2$. Plug these values into the equation $y = -\frac{3}{2}x + b$:

$-2 = -\frac{3}{2}(0) + b$

Simplifying: $-2 = b$

So, the y-intercept is $b = -2$.

Final equation:

Now, substitute $b = -2$ and $m = -\frac{3}{2}$ into the slope-intercept form:

$y = -\frac{3}{2}x - 2$

This is the equation of the line.

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Would you like further details on any step? Here are some related questions for practice:

1. Find the equation of a line passing through points $(3, 4)$ and $(6, 7)$.
2. What is the slope of a line parallel to the line $y = \frac{5}{3}x + 1$?
3. Determine the equation of a line with slope $2$ and passing through the point $(1, -3)$.
4. How do you find the equation of a line given its slope and one point?
5. Find the slope of a line passing through $(4, 8)$ and $(7, 14)$.

Tip: The slope of a line remains the same for all points along that line. This makes the slope a very useful property for determining equations of lines.