To find the equation of the line that is perpendicular to the line given by the equation \( y = x - 3 \) and passes through the point \( (-6, 0) \), we will follow these steps:
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Determine the slope of the given line: The equation \( y = x - 3 \) can be rewritten in slope-intercept form \( y = mx + b \), where \( m \) is the slope. The slope \( m \) of this line is \( 1 \).
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Find the slope of the perpendicular line: The slope of a line that is perpendicular to another is the negative reciprocal of the original line's slope. Therefore, if the slope of the original line is \( 1 \), the slope \( m_{\perp} \) of the perpendicular line will be: \[ m_{\perp} = -\frac{1}{1} = -1. \]
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Use the point-slope form of the line: We can now use the point-slope form of the equation of a line, which is given by: \[ y - y_1 = m(x - x_1), \] where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. We have the point \( (-6, 0) \) and the slope \( m_{\perp} = -1 \): \[ y - 0 = -1(x - (-6)). \] Simplifying this: \[ y = -1(x + 6). \] \[ y = -x - 6. \]
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Final equation: The equation of the line perpendicular to \( y = x - 3 \) that passes through \( (-6, 0) \) is: \[ y = -x - 6. \]
This can also be written in standard form \( x + y + 6 = 0 \) if desired.
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