To find the equation of a line perpendicular to [tex]\(5x + 8y = 9\)[/tex] that passes through the point [tex]\((6, -7)\)[/tex], we can follow these steps:
1. Find the slope of the given line: The given line's equation is [tex]\(5x + 8y = 9\)[/tex]. To find the slope, we need to rewrite this equation in the slope-intercept form [tex]\(y = mx + b\)[/tex].
[tex]\[
8y = -5x + 9
\][/tex]
[tex]\[
y = \left(-\frac{5}{8}\right)x + \frac{9}{8}
\][/tex]
Thus, the slope [tex]\(m\)[/tex] of the given line is [tex]\(-\frac{5}{8}\)[/tex].
2. Find the slope of the perpendicular line: The slope of a line perpendicular to another is the negative reciprocal of the slope of the original line. So, the slope of the perpendicular line is:
[tex]\[
\text{slope}_{\text{perpendicular}} = -\frac{1}{-\frac{5}{8}} = \frac{8}{5}
\][/tex]
3. Write the equation of the perpendicular line: The line we seek passes through the point [tex]\((6, -7)\)[/tex] with a slope of [tex]\(\frac{8}{5}\)[/tex]. We can use the point-slope form of the equation:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\( (x_1, y_1) = (6, -7) \)[/tex] and [tex]\( m = \frac{8}{5} \)[/tex].
[tex]\[
y - (-7) = \frac{8}{5}(x - 6)
\][/tex]
[tex]\[
y + 7 = \frac{8}{5}(x - 6)
\][/tex]
4. Simplify the equation to slope-intercept form (if desired):
[tex]\[
y + 7 = \frac{8}{5}x - \frac{8 \cdot 6}{5}
\][/tex]
[tex]\[
y + 7 = \frac{8}{5}x - \frac{48}{5}
\][/tex]
[tex]\[
y = \frac{8}{5}x - \frac{48}{5} - 7
\][/tex]
[tex]\[
y = \frac{8}{5}x - \frac{48}{5} - \frac{35}{5}
\][/tex]
[tex]\[
y = \frac{8}{5}x - \frac{83}{5}
\][/tex]
Therefore, the equation of the line perpendicular to [tex]\( 5x + 8y = 9 \)[/tex] and passing through the point [tex]\( (6, -7) \)[/tex] is:
[tex]\[
y = \frac{8}{5}x - \frac{83}{5}
\][/tex]
