To derive the equation of the line in slope-intercept form \(y = mx + b\), we first need to determine the slope (\(m\)) using the two given points: \((0, -2)\) and \((1, 3)\).
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Now, substituting our points into this formula:
- Let \((x_1, y_1) = (0, -2)\)
- Let \((x_2, y_2) = (1, 3)\)
Calculating the slope:
\[ m = \frac{3 - (-2)}{1 - 0} = \frac{3 + 2}{1} = \frac{5}{1} = 5 \]
Now that we have the slope \(m = 5\), we can use one of the points to find the y-intercept \(b\). Since one of the points is \((0, -2)\), we can directly see that when \(x = 0\), \(y = -2\). Thus, the y-intercept \(b = -2\).
Therefore, the equation of the line in slope-intercept form is:
\[ y = 5x - 2 \]