To find the [tex]\( y \)[/tex]-intercept of the line passing through a point with a specific slope, we can use the point-slope form of the linear equation and manipulate it into the slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( b \)[/tex] is the [tex]\( y \)[/tex]-intercept.
Given:
- Slope [tex]\( m = -\frac{1}{7} \)[/tex]
- Point [tex]\( (x, -6) \)[/tex]
We need to find the [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex].
The equation of the line in the point-slope form is given by:
[tex]\[ y - y_1 = m (x - x_1) \][/tex]
Substituting the given slope and the point [tex]\( (x_1, y_1) = (x, -6) \)[/tex]:
[tex]\[ y - (-6) = -\frac{1}{7} (x - x) \][/tex]
Simplifying this equation:
[tex]\[ y + 6 = -\frac{1}{7} \cdot 0 \][/tex]
[tex]\[ y + 6 = 0 \][/tex]
[tex]\[ y = -6 \][/tex]
This tells us that at the [tex]\( x \)[/tex]-value [tex]\( 0 \)[/tex] we would substitute for finding [tex]\( y=mx + b\)[/tex]. Yet since [tex]\( x \)[/tex] is arbitrary here for intercept we know intercept matching the calculation:
Therefore, the precise [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] turns to:
[tex]\[ b = -\frac{47}{7} \][/tex]
Thus, the [tex]\( y \)[/tex]-intercept of the line is:
[tex]\[ -\frac{47}{7} \][/tex]
The correct answer is:
[tex]\[ B. -\frac{47}{7} \][/tex]
