To derive the equation of the line in slope-intercept form (y = mx + b), we need to find the slope (m) and the y-intercept (b) using the two points provided: (3, 0) and (0, -2).
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Finding the slope (m): The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, we can assign:
- \((x_1, y_1) = (3, 0)\)
- \((x_2, y_2) = (0, -2)\)
Plugging in the values: \[ m = \frac{-2 - 0}{0 - 3} = \frac{-2}{-3} = \frac{2}{3} \]
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Finding the y-intercept (b): The y-intercept \(b\) is the value of \(y\) when \(x = 0\). From the second point (0, -2), we see that \(b = -2\).
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Constructing the equation: Now that we have the slope \(m = \frac{2}{3}\) and the y-intercept \(b = -2\), we can write the equation of the line in slope-intercept form: \[ y = \frac{2}{3}x - 2 \]
Thus, the equation of the line in slope-intercept form is: \[ y = \frac{2}{3}x - 2 \]