Sure, let's go through the solution step-by-step to find the equation of the line containing the point [tex]\((8, 9)\)[/tex] and perpendicular to the given line [tex]\(6x + y = 8\)[/tex].
1. Rewrite the given line in slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[6x + y = 8\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[y = -6x + 8\][/tex]
So, the slope [tex]\(m_1\)[/tex] of the given line is [tex]\(-6\)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the original slope. The slope [tex]\(m_2\)[/tex] of the line we are looking for is:
[tex]\[
m_2 = \frac{1}{-m_1} = \frac{1}{-(-6)} = \frac{1}{6}
\][/tex]
3. Use the point-slope form to write the equation:
The point-slope form of a line equation is:
[tex]\[
y - y_1 = m_2(x - x_1)
\][/tex]
Plugging in the given point [tex]\((8, 9)\)[/tex] and the slope [tex]\(m_2 = \frac{1}{6}\)[/tex]:
[tex]\[
y - 9 = \frac{1}{6}(x - 8)
\][/tex]
4. Simplify to get the slope-intercept form [tex]\(y = mx + b\)[/tex]:
[tex]\[
y - 9 = \frac{1}{6}x - \frac{8}{6}
\][/tex]
Simplify [tex]\(\frac{8}{6}\)[/tex] to [tex]\(\frac{4}{3}\)[/tex]:
[tex]\[
y - 9 = \frac{1}{6}x - \frac{4}{3}
\][/tex]
Adding 9 to both sides to solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{1}{6}x - \frac{4}{3} + 9
\][/tex]
Convert the integer 9 to a fraction with a common denominator:
[tex]\[
9 = \frac{27}{3}
\][/tex]
Continue simplifying:
[tex]\[
y = \frac{1}{6}x - \frac{4}{3} + \frac{27}{3}
\][/tex]
[tex]\[
y = \frac{1}{6}x + \frac{23}{3}
\][/tex]
So, the equation of the line containing the point [tex]\((8, 9)\)[/tex] and perpendicular to the line [tex]\(6x + y = 8\)[/tex] is:
[tex]\[
y = \frac{1}{6}x + \frac{23}{3}
\][/tex]
