To find the initial value (the y-intercept) of a linear relationship using the given points (3, 42.25) and (5, 50.75), we can follow these steps:
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Determine the Slope (m): The first step is to calculate the slope of the line, which is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (3, 42.25) as \((x_1, y_1)\) and (5, 50.75) as \((x_2, y_2)\):
\[ m = \frac{50.75 - 42.25}{5 - 3} = \frac{8.50}{2} = 4.25 \]
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Use the Point-Slope Form of the Linear Equation: The point-slope form of the linear equation is:
\[ y - y_1 = m(x - x_1) \]
Using the point (3, 42.25):
\[ y - 42.25 = 4.25(x - 3) \]
Expand this equation:
\[ y - 42.25 = 4.25x - 12.75 \] \[ y = 4.25x + 29.50 \]
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Identify the Initial Value (y-intercept): The y-intercept (initial value) is the value of \(y\) when \(x = 0\). From the equation \(y = 4.25x + 29.50\), the y-intercept is 29.50.
Thus, to find the initial value for this linear relationship, you can use the formula for linear equations to derive the equation of the line and look at the y-intercept. In this case, the initial value is 29.50.
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