To write the equation of a line that is parallel to the given line \( y = -5x + 3 \) and passes through the point \((-6, 3)\), we need to follow these steps:
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Identify the slope of the given line.
The equation \( y = -5x + 3 \) is in slope-intercept form, \( y = mx + b \), where \( m \) is the slope. Here, the slope \( m \) is \(-5\). -
Use the same slope for the parallel line.
Since parallel lines have the same slope, our new line will also have a slope of \(-5\). -
Use the point-slope form of a line equation.
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 the point the line passes through. Here, \((x_1, y_1) = (-6, 3)\) and \( m = -5 \). -
Substitute the values into the point-slope form: \[ y - 3 = -5(x - (-6)) \] \[ y - 3 = -5(x + 6) \]
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Distribute and simplify: \[ y - 3 = -5x - 30 \] \[ y = -5x - 30 + 3 \] \[ y = -5x - 27 \]
Thus, the equation of the line that is parallel to the given line and passes through the point \((-6, 3)\) is: \[ \boxed{y = -5x - 27} \]