To derive the equation of the line in slope-intercept form \(y = mx + b\), you need to determine the slope \(m\) and the y-intercept \(b\).
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Calculate the slope (\(m\)): The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points \((3, 1)\) and \((0, -1)\): \[ (x_1, y_1) = (3, 1) \quad \text{and} \quad (x_2, y_2) = (0, -1) \] Plugging in the coordinates: \[ m = \frac{-1 - 1}{0 - 3} = \frac{-2}{-3} = \frac{2}{3} \]
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Find the y-intercept (\(b\)): The y-intercept is the value of \(y\) when \(x = 0\). From the point \((0, -1)\), we can see that the y-intercept \(b = -1\).
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Write the equation in slope-intercept form: Now that we have \(m = \frac{2}{3}\) and \(b = -1\), we can write the equation of the line: \[ y = \frac{2}{3}x - 1 \]
So, the equation of the line in slope-intercept form is: \[ y = \frac{2}{3}x - 1 \]
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