To find the initial value of a linear relationship given the ordered pairs (3, 42.25) and (5, 50.75), first determine the slope (m) using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \( (x_1, y_1) = (3, 42.25) \) and \( (x_2, y_2) = (5, 50.75) \):
\[ m = \frac{50.75 - 42.25}{5 - 3} = \frac{8.5}{2} = 4.25 \]
Next, use the point-slope form of the linear equation \( y - y_1 = m(x - x_1) \) with one of the points to find the equation. Using (3, 42.25):
\[ y - 42.25 = 4.25(x - 3) \]
To find the initial value (y-intercept), set \( x = 0 \) and solve for \( y \):
\[ y - 42.25 = 4.25(0 - 3) \]
\[ y - 42.25 = -12.75 \]
\[ y = 42.25 - 12.75 = 29.5 \]
Thus, the initial value (y-intercept) is 29.5.
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