To find the slope [tex]\( k \)[/tex] of the line [tex]\( y = kx + 4 \)[/tex] when the line passes through the point [tex]\((c, d)\)[/tex], we need to follow these steps:
1. Recall the Equation of the Line: The line is described by the equation
[tex]\[
y = kx + 4
\][/tex]
Here, [tex]\( k \)[/tex] is the slope of the line, and [tex]\( 4 \)[/tex] is the y-intercept.
2. Point-Slope Formula: Since the point [tex]\((c, d)\)[/tex] lies on the line, this point must satisfy the line equation. Substitute [tex]\( x = c \)[/tex] and [tex]\( y = d \)[/tex] into the line equation:
[tex]\[
d = kc + 4
\][/tex]
3. Solving for the Slope [tex]\( k \)[/tex]: We need to isolate [tex]\( k \)[/tex] in the equation. First, subtract 4 from both sides of the equation:
[tex]\[
d - 4 = kc
\][/tex]
4. Divide by [tex]\( c \)[/tex]: Finally, divide both sides by [tex]\( c \)[/tex] to solve for [tex]\( k \)[/tex]:
[tex]\[
k = \frac{d - 4}{c}
\][/tex]
Therefore, 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]
