To determine the value of [tex]\( b \)[/tex] (the y-intercept) in the equation of the line that passes through points [tex]\( J(-4, -5) \)[/tex] and [tex]\( K(-6, 3) \)[/tex], we proceed as follows:
1. Find the slope (m) of the line:
The formula for the slope between two points [tex]\( J(x1, y1) \)[/tex] and [tex]\( K(x2, y2) \)[/tex] is:
[tex]\[
m = \frac{y2 - y1}{x2 - x1}
\][/tex]
For points [tex]\( J(-4, -5) \)[/tex] and [tex]\( K(-6, 3) \)[/tex]:
[tex]\[
m = \frac{3 - (-5)}{-6 - (-4)} = \frac{3 + 5}{-6 + 4} = \frac{8}{-2} = -4
\][/tex]
2. Use the slope-intercept form (y = mx + b) to find the y-intercept (b):
The slope-intercept form of the equation is:
[tex]\[
y = mx + b
\][/tex]
Substitute one of the points and the calculated slope into the equation to solve for [tex]\( b \)[/tex]. Using point [tex]\( J(-4, -5) \)[/tex]:
[tex]\[
-5 = -4(-4) + b
\][/tex]
Simplify and solve for [tex]\( b \)[/tex]:
[tex]\[
-5 = 16 + b
\][/tex]
[tex]\[
b = -5 - 16
\][/tex]
[tex]\[
b = -21
\][/tex]
So, the y-intercept [tex]\( b \)[/tex] for the line passing through points [tex]\( J(-4,-5) \)[/tex] and [tex]\( K(-6,3) \)[/tex] is [tex]\(-21\)[/tex].
Therefore, the correct answer is:
[tex]\[
-21
\][/tex]
