To find the equation of a line parallel to \(5x + y = 6\), we first need to determine the slope of the given line. The equation is in the standard form \(Ax + By = C\).
- Rearranging to slope-intercept form \(y = mx + b\): \[ y = -5x + 6 \] From this, we can see that the slope \(m\) is \(-5\).
Since parallel lines share the same slope, the slope of the new line will also be \(-5\). Now we use the point-slope form of the equation of the line, which is given by: \[ y - y_1 = m(x - x_1) \] Where \((x_1, y_1)\) is the point through which the line passes (in this case, \((6, 1)\)), and \(m\) is the slope.
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Substituting \(m = -5\), \(x_1 = 6\), and \(y_1 = 1\): \[ y - 1 = -5(x - 6) \]
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Distributing the \(-5\): \[ y - 1 = -5x + 30 \]
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Adding \(1\) to both sides: \[ y = -5x + 31 \]
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To convert this to standard form \(Ax + By = C\), we can rearrange it: \[ 5x + y = 31 \]
Thus, the equation of the line parallel to \(5x + y = 6\) that passes through the point \((6, 1)\) is:
\[ \boxed{5x + y = 31} \]
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