To determine the equation of the line that passes through the point [tex]\((-9, 1)\)[/tex] and is parallel to the line given by the equation [tex]\(8x - 9y - 5 = 0\)[/tex], we first need to find the slope of the given line.
1. Finding the Slope:
- The given line is [tex]\(8x - 9y - 5 = 0\)[/tex].
- To find the slope, we'll rewrite this equation in slope-intercept form [tex]\((y = mx + b)\)[/tex], where [tex]\(m\)[/tex] represents the slope.
- Starting from [tex]\(8x - 9y - 5 = 0\)[/tex], solve for [tex]\(y\)[/tex]:
[tex]\[
8x - 9y = 5 \quad \text{(adding 5 to both sides)}
\][/tex]
[tex]\[
-9y = -8x + 5 \quad \text{(subtracting 8x from both sides)}
\][/tex]
[tex]\[
y = \frac{8}{9}x - \frac{5}{9} \quad \text{(dividing by -9)}
\][/tex]
- The slope [tex]\(m\)[/tex] of the given line is [tex]\(\frac{8}{9}\)[/tex].
2. Equation in Point-Slope Form:
- The point-slope form of the equation of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the given point and [tex]\(m\)[/tex] is the slope.
- Substituting the point [tex]\((-9, 1)\)[/tex] and the slope [tex]\(\frac{8}{9}\)[/tex] into the equation, we get:
[tex]\[
y - 1 = \frac{8}{9}(x + 9)
\][/tex]
- The equation of the line in point-slope form is:
[tex]\[
y - 1 = \frac{8}{9}(x - (-9))
\][/tex]
Thus, the equation of the line in point-slope form is [tex]\(\boxed{y - 1 = \frac{8}{9}(x + 9)}\)[/tex].
