To find the initial value of a linear relationship given two ordered pairs (x₁, y₁) and (x₂, y₂), follow these steps:
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Calculate the slope (m) using the formula: \[ m = \frac{y₂ - y₁}{x₂ - x₁} \] For the pairs (3, 42.25) and (5, 50.75): \[ m = \frac{50.75 - 42.25}{5 - 3} = \frac{8.5}{2} = 4.25 \]
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Use the point-slope form of the linear equation: \[ y - y₁ = m(x - x₁) \] Choose one of the points, for example, (3, 42.25): \[ y - 42.25 = 4.25(x - 3) \]
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Convert to slope-intercept form (y = mx + b) to find the initial value (b): \[ y = 4.25x - 12.75 + 42.25 \implies y = 4.25x + 29.5 \]
Thus, the initial value (y-intercept) is 29.5.
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