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14. The linear function [tex]f[/tex] is defined by [tex]f(x) = c x + d[/tex], where [tex]c[/tex] and [tex]d[/tex] are constants.

If [tex]f(50) = 27,000[/tex] and [tex]f(100) = 38,000[/tex], what is the value of [tex]c[/tex]?


Sagot :

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].
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