To derive the equation of the line in slope-intercept form (which is \( y = mx + b \)), we first need to determine the slope \( m \) using the two given points: \( (1, 8) \) and \( (0, 4) \).
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of the points:
- \( (x_1, y_1) = (1, 8) \)
- \( (x_2, y_2) = (0, 4) \)
Now we calculate \( m \):
\[ m = \frac{4 - 8}{0 - 1} = \frac{-4}{-1} = 4 \]
So, the slope \( m \) is 4.
Next, we can use one of the points to find the y-intercept \( b \). We can use the point \( (0, 4) \), which is already in the form where \( x = 0 \) indicates the y-intercept.
For the point \( (0, 4) \):
\[ y = mx + b \implies 4 = 4(0) + b \implies b = 4 \]
Now, we have both the slope \( m \) and the y-intercept \( b \):
- Slope \( m = 4 \)
- Y-intercept \( b = 4 \)
Putting it all together, the equation of the line in slope-intercept form is:
\[ y = 4x + 4 \]
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