Question
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
Upstudy AI Solution
Tutor-Verified Answer
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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Simplify this solution Mind Expander
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 \( \frac{5}{7} \).
