To find the equation of a line with a given slope and a point, we can use the point-slope form of the equation of a line, which is:
\[ y - y_1 = m(x - x_1) \]
where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope.
Given the point \((-2, 4)\) and the slope \(m = \frac{2}{5}\), we can substitute these values into the equation:
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Set \(x_1 = -2\) and \(y_1 = 4\): \[ y - 4 = \frac{2}{5}(x - (-2)) \]
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This simplifies to: \[ y - 4 = \frac{2}{5}(x + 2) \]
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Distributing the slope: \[ y - 4 = \frac{2}{5}x + \frac{4}{5} \]
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Now, add 4 (or \( \frac{20}{5}\)) to both sides to isolate \(y\): \[ y = \frac{2}{5}x + \frac{4}{5} + \frac{20}{5} \] \[ y = \frac{2}{5}x + \frac{24}{5} \]
Therefore, the equation of the line that passes through \((-2, 4)\) with a slope of \(\frac{2}{5}\) is:
\[ y = \frac{2}{5}x + \frac{24}{5} \]