To derive the equation of the line in slope-intercept form \(y = mx + b\) given the points \((0, 2)\) and \((1, -1)\), follow these steps:
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Calculate the slope (m): The slope \(m\) of a line passing through the 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, \((x_1, y_1) = (0, 2)\) and \((x_2, y_2) = (1, -1)\): \[ m = \frac{-1 - 2}{1 - 0} = \frac{-3}{1} = -3 \]
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Use the slope and a point to find the y-intercept (b): We can use one of the points to find \(b\). Since we already have the point \((0, 2)\), which is the y-intercept (where \(x = 0\)), we can immediately see that: \[ b = 2 \]
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Write the slope-intercept form: Now that we have \(m\) and \(b\): \[ y = mx + b \] Substituting in our values: \[ y = -3x + 2 \]
So, the equation of the line in slope-intercept form is: \[ \boxed{y = -3x + 2} \]