To find the equation of a line that is perpendicular to the line described by the equation \( y - 7 = -x \) and passes through the point \( (9, 8) \), we can follow these steps:
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Convert the given equation to slope-intercept form: The equation \( y - 7 = -x \) can be rearranged to find the slope: \[ y = -x + 7 \] Here, the slope \( m_1 \) of the line is \( -1 \).
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Find the slope of the perpendicular line: The slope of a line perpendicular to another is the negative reciprocal of the original slope. Since the slope of the original line is \( -1 \), the slope of the perpendicular line \( m_2 \) will be: \[ m_2 = -\frac{1}{-1} = 1 \]
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Use the point-slope form to find the equation: We can use the point-slope formula \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point the line passes through, and \( m \) is the slope of the line. Plugging in the point \( (9, 8) \) and \( m = 1 \): \[ y - 8 = 1(x - 9) \]
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Simplifying the equation: \[ y - 8 = x - 9 \] \[ y = x - 1 \]
Thus, the equation of the line that is perpendicular to \( y - 7 = -x \) and passes through the point \( (9, 8) \) is: \[ \boxed{y = x - 1} \]