To determine the value of [tex]\( c \)[/tex] in the linear function [tex]\( f(x) = cx + d \)[/tex], given the values [tex]\( f(50) = 27,000 \)[/tex] and [tex]\( f(100) = 38,000 \)[/tex], we can set up a system of linear equations based on these points.
1. Setting up the equations:
Using the given function [tex]\( f(x) = cx + d \)[/tex]:
- For [tex]\( x = 50 \)[/tex]:
[tex]\[
f(50) = 50c + d = 27,000 \quad \text{(Equation 1)}
\][/tex]
- For [tex]\( x = 100 \)[/tex]:
[tex]\[
f(100) = 100c + d = 38,000 \quad \text{(Equation 2)}
\][/tex]
2. Solving the system of equations:
We have the following system of linear equations:
[tex]\[
\begin{cases}
50c + d = 27,000 \\
100c + d = 38,000
\end{cases}
\][/tex]
To eliminate [tex]\( d \)[/tex], we subtract Equation 1 from Equation 2:
[tex]\[
(100c + d) - (50c + d) = 38,000 - 27,000
\][/tex]
This simplifies to:
[tex]\[
50c = 11,000
\][/tex]
Now, solve for [tex]\( c \)[/tex] by dividing both sides of the equation by 50:
[tex]\[
c = \frac{11,000}{50}
\][/tex]
3. Simplifying the result:
Calculate the division:
[tex]\[
c = 220.0
\][/tex]
Hence, the value of [tex]\( c \)[/tex] is [tex]\( \boxed{220.0} \)[/tex].
