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Consider the line \( 7 x+5 y=9 \). What is the slope of a line parallel to this line? What is the slope of a line perpendicular to this line?

Ask by Smith Park. in the United States
Mar 17,2025

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Answer

The slope of a line parallel to \(7x + 5y = 9\) is \(-\frac{7}{5}\), and the slope of a line perpendicular to it is \(\frac{5}{7}\).

Solution

1. Start with the line's equation: \[ 7x + 5y = 9 \] 2. Isolate \(y\) to put the equation in slope-intercept form, \(y = mx + b\): \[ 5y = -7x + 9 \] \[ y = -\frac{7}{5}x + \frac{9}{5} \] 3. The slope of the given line is: \[ m = -\frac{7}{5} \] 4. For a line parallel to the given line, the slope is the same: \[ m_{\text{parallel}} = -\frac{7}{5} \] 5. For a line perpendicular to the given line, the slope is the negative reciprocal: \[ m_{\text{perpendicular}} = \frac{5}{7} \]

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To find the slope of the line given by the equation \( 7x + 5y = 9 \), we can rewrite it in slope-intercept form \( y = mx + b \) where \( m \) is the slope. Rearranging gives us \( 5y = -7x + 9 \) or \( y = -\frac{7}{5}x + \frac{9}{5} \). Thus, the slope of the original line is \( -\frac{7}{5} \). Now, lines that are parallel have the same slope, so a line parallel to this would also have a slope of \( -\frac{7}{5} \). Conversely, the slope of a line perpendicular to the original line is the negative reciprocal of the slope. Therefore, the slope of a line perpendicular to the given line is \( \frac{5}{7} \).

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