To find the equation of a line that is parallel to the line \( y = \frac{2}{5}x + 1 \) and passes through the point \( (5, 5) \), we follow these steps:
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Identify the slope of the original line: The slope of the line \( y = \frac{2}{5}x + 1 \) is \( \frac{2}{5} \). Since parallel lines have the same slope, the slope of the new line will also be \( \frac{2}{5} \).
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Use the point-slope form of a line: The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. In this case, \( m = \frac{2}{5} \) and the point is \( (5, 5) \).
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Plug in the values: \[ y - 5 = \frac{2}{5}(x - 5) \]
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Simplify the equation: Distributing the slope on the right side gives: \[ y - 5 = \frac{2}{5}x - \frac{2}{5} \cdot 5 \] \[ y - 5 = \frac{2}{5}x - 2 \] Now, add 5 to both sides: \[ y = \frac{2}{5}x - 2 + 5 \] \[ y = \frac{2}{5}x + 3 \]
So, the equation of the line that is parallel to \( y = \frac{2}{5}x + 1 \) and passes through the point \( (5, 5) \) is: \[ \boxed{y = \frac{2}{5}x + 3} \]