Certainly! Let's walk through the problem step-by-step to find the value of [tex]\( k \)[/tex].
1. Understand the Problem: We are given the equation of a line [tex]\( y = kx + 4 \)[/tex]. We also know that the point [tex]\((c, d)\)[/tex] lies on this line. Our goal is to express the slope [tex]\( k \)[/tex] in terms of [tex]\( c \)[/tex] and [tex]\( d \)[/tex].
2. Point-Slope Relationship: Since the point [tex]\((c, d)\)[/tex] lies on the line [tex]\( y = kx + 4 \)[/tex], we know that when [tex]\( x = c \)[/tex], [tex]\( y = d \)[/tex]. This gives us the relationship to work with:
[tex]\[
d = kc + 4
\][/tex]
3. Solving for [tex]\( k \)[/tex]: We need to isolate [tex]\( k \)[/tex] in the equation. Start by subtracting 4 from both sides of the equation:
[tex]\[
d - 4 = kc
\][/tex]
4. Isolate [tex]\( k \)[/tex]: Now solve for [tex]\( k \)[/tex] by dividing both sides by [tex]\( c \)[/tex]:
[tex]\[
k = \frac{d - 4}{c}
\][/tex]
So the slope [tex]\( k \)[/tex] of the line in terms of [tex]\( c \)[/tex] and [tex]\( d \)[/tex] is:
[tex]\[
k = \frac{d - 4}{c}
\][/tex]
This is the required expression for [tex]\( k \)[/tex].
