To find the equation of a line parallel to \(5x + y = 6\) that passes through the point \( (15, -4) \), we start by identifying the slope of the given equation.
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Find the slope of the original line: Rearranging \(5x + y = 6\) into slope-intercept form \(y = mx + b\): \[ y = -5x + 6 \] The slope \(m\) of this line is \(-5\).
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Use the same slope for the new line: Since parallel lines have the same slope, the slope of the desired line is also \(-5\).
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Use the point-slope form of the line equation: The point-slope form is given by: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) = (15, -4) \) and \(m = -5\): \[ y - (-4) = -5(x - 15) \] Simplifying this: \[ y + 4 = -5x + 75 \] \[ y = -5x + 71 \]
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Convert to standard form: To convert \(y = -5x + 71\) to standard form \(Ax + By = C\), we rearrange it: \[ 5x + y = 71 \]
The final answer, in standard form, is: \[ \boxed{5x + y = 71} \]
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