To derive the equation of the line given the points \((0, 4)\) and \((1, 8)\), we can follow these steps:
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Calculate the slope (m): The slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \((x_1, y_1) = (0, 4)\) and \((x_2, y_2) = (1, 8)\). Plugging in the values: \[ m = \frac{8 - 4}{1 - 0} = \frac{4}{1} = 4 \]
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Use the point-slope form of the line: The point-slope form of the line is given by: \[ y - y_1 = m(x - x_1) \] We'll use one of the points, say \((0, 4)\), and the slope \(m = 4\): \[ y - 4 = 4(x - 0) \]
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Simplify to slope-intercept form (y = mx + b): Distributing the slope: \[ y - 4 = 4x \] Adding \(4\) to both sides gives: \[ y = 4x + 4 \]
So, the equation of the line in slope-intercept form is: \[ y = 4x + 4 \]