To solve this problem, we are dealing with the linear equation of the form [tex]\( f(x) = mx + c \)[/tex]. We need to determine the values of [tex]\( m \)[/tex] and [tex]\( c \)[/tex] given two points on the line: [tex]\( (3, 4) \)[/tex] and [tex]\( (6, -2) \)[/tex]. Additionally, we need to find the value of [tex]\( mac \)[/tex] and [tex]\( f(n) \)[/tex].
### Step-by-Step Solution
1. Set up the equations:
Since [tex]\( f(x) = mx + c \)[/tex], we substitute the given points into the equation.
For the point [tex]\( (3, 4) \)[/tex]:
[tex]\[ 4 = 3m + c \][/tex]
For the point [tex]\( (6, -2) \)[/tex]:
[tex]\[ -2 = 6m + c \][/tex]
2. Form the system of equations:
We have two equations:
[tex]\[
\begin{cases}
4 = 3m + c & \quad \text{(Equation 1)} \\
-2 = 6m + c & \quad \text{(Equation 2)}
\end{cases}
\][/tex]
3. Solve the system of equations:
Subtract Equation 1 from Equation 2 to eliminate [tex]\( c \)[/tex]:
[tex]\[
(-2) - 4 = (6m + c) - (3m + c)
\][/tex]
Simplify:
[tex]\[
-6 = 3m
\][/tex]
Solving for [tex]\( m \)[/tex]:
[tex]\[
m = -2
\][/tex]
4. Find [tex]\( c \)[/tex]:
Substitute [tex]\( m = -2 \)[/tex] back into Equation 1:
[tex]\[
4 = 3(-2) + c
\][/tex]
Simplify:
[tex]\[
4 = -6 + c
\][/tex]
Solving for [tex]\( c \)[/tex]:
[tex]\[
c = 10
\][/tex]
5. Calculate [tex]\( mac \)[/tex]:
Given that [tex]\( m = -2 \)[/tex] and [tex]\( c = 10 \)[/tex]:
[tex]\[
mac = m \cdot a \cdot c
\][/tex]
Assuming [tex]\( a = 1 \)[/tex] for simplicity as typical problems set [tex]\( a = 1 \)[/tex] if not specified otherwise, we get:
[tex]\[
mac = (-2) \cdot 1 \cdot 10 = -20
\][/tex]
6. Find [tex]\( f(n) \)[/tex]:
The function [tex]\( f(x) \)[/tex] is now completely determined:
[tex]\[
f(x) = -2x + 10
\][/tex]
To find [tex]\( f(n) \)[/tex]:
[tex]\[
f(n) = -2n + 10
\][/tex]
### Summary of Results
- [tex]\( m = -2 \)[/tex]
- [tex]\( c = 10 \)[/tex]
- [tex]\( mac = -20 \)[/tex]
- [tex]\( f(n) = -2n + 10 \)[/tex]
Therefore, the value of [tex]\( mac \)[/tex] is [tex]\( -20 \)[/tex], and the function [tex]\( f(n) \)[/tex] simplifies to [tex]\( f(n) = -2n + 10 \)[/tex].
