To derive the equation of the line that passes through the points (0, 1) and (1, -4), we can follow these steps:
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Identify the points:
- Point 1: \((0, 1)\)
- Point 2: \((1, -4)\)
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Calculate the slope (m): The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the coordinates of the points:
- \(x_1 = 0\), \(y_1 = 1\)
- \(x_2 = 1\), \(y_2 = -4\)
\[ m = \frac{-4 - 1}{1 - 0} = \frac{-5}{1} = -5 \]
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Use the slope-intercept form: The slope-intercept form of a line is given by: \[ y = mx + b \] where \(m\) is the slope and \(b\) is the y-intercept. We already know the slope \(m = -5\).
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Find the y-intercept (b): Since we have the point (0, 1), we can directly identify the y-intercept from this point. When \(x = 0\), \(y = 1\), which means: \[ b = 1 \]
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Write the equation: Substituting the values of \(m\) and \(b\) into the slope-intercept form: \[ y = -5x + 1 \]
Thus, the equation of the line in slope-intercept form is: \[ \boxed{y = -5x + 1} \]
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