To determine the y-intercept ([tex]\(b\)[/tex]) of a line passing through the point [tex]\((9, -7)\)[/tex] with a slope of [tex]\(-\frac{1}{8}\)[/tex], we can use the point-slope form of the line equation, which is:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\(m\)[/tex] is the slope of the line,
- [tex]\((x_1, y_1)\)[/tex] is a point on the line,
- [tex]\(b\)[/tex] is the y-intercept we want to find.
Substitute the given values into the equation. Here, [tex]\(x_1 = 9\)[/tex], [tex]\(y_1 = -7\)[/tex], and [tex]\(m = -\frac{1}{8}\)[/tex]:
[tex]\[ y_1 = m \cdot x_1 + b \][/tex]
[tex]\[ -7 = -\frac{1}{8} \cdot 9 + b \][/tex]
First, calculate the product of the slope and the x-coordinate of the point:
[tex]\[ -\frac{1}{8} \cdot 9 = - \frac{9}{8} \][/tex]
Now, substitute this value back into the equation:
[tex]\[ -7 = - \frac{9}{8} + b \][/tex]
To isolate [tex]\(b\)[/tex], add [tex]\(\frac{9}{8}\)[/tex] to both sides of the equation:
[tex]\[ -7 + \frac{9}{8} = b \][/tex]
Convert [tex]\(-7\)[/tex] to a fraction with a denominator of 8 to facilitate addition:
[tex]\[ -7 = -\frac{56}{8} \][/tex]
Now perform the addition:
[tex]\[ -\frac{56}{8} + \frac{9}{8} = b \][/tex]
[tex]\[ -\frac{56 - 9}{8} = b \][/tex]
[tex]\[ -\frac{47}{8} = b \][/tex]
Thus, the y-intercept [tex]\(b\)[/tex] for the line passing through the point [tex]\((9, -7)\)[/tex] with a slope of [tex]\(-\frac{1}{8}\)[/tex] is [tex]\(-\frac{47}{8}\)[/tex].
Therefore, the correct answer is:
A. [tex]\(-\frac{47}{8}\)[/tex]
