To solve for [tex]\(m\)[/tex] and [tex]\(c\)[/tex] given the linear function [tex]\(f(x) = mx + c\)[/tex] with the points [tex]\(f(4) = 11\)[/tex] and [tex]\(f(5) = 13\)[/tex], follow these steps:
1. Set up the equations using the given points:
We know that if [tex]\(f(x) = mx + c\)[/tex], then plugging in the values for the points should satisfy the equation.
- For [tex]\(x = 4\)[/tex] and [tex]\(f(4) = 11\)[/tex]:
[tex]\[
m \cdot 4 + c = 11
\][/tex]
- For [tex]\(x = 5\)[/tex] and [tex]\(f(5) = 13\)[/tex]:
[tex]\[
m \cdot 5 + c = 13
\][/tex]
2. Write the system of linear equations:
From the points [tex]\(f(4) = 11\)[/tex] and [tex]\(f(5) = 13\)[/tex]:
[tex]\[
4m + c = 11 \quad \text{(1)}
\][/tex]
[tex]\[
5m + c = 13 \quad \text{(2)}
\][/tex]
3. Solve the system of equations:
First, subtract equation (1) from equation (2) to eliminate [tex]\(c\)[/tex]:
[tex]\[
(5m + c) - (4m + c) = 13 - 11
\][/tex]
Simplifying this:
[tex]\[
5m + c - 4m - c = 2
\][/tex]
[tex]\[
m = 2
\][/tex]
4. Find the value of [tex]\(c\)[/tex]:
Substitute [tex]\(m = 2\)[/tex] back into one of the original equations. Using equation (1):
[tex]\[
4m + c = 11
\][/tex]
Substitute [tex]\(m = 2\)[/tex]:
[tex]\[
4 \cdot 2 + c = 11
\][/tex]
[tex]\[
8 + c = 11
\][/tex]
Subtract 8 from both sides:
[tex]\[
c = 3
\][/tex]
Hence, the values of [tex]\(m\)[/tex] and [tex]\(c\)[/tex] are:
[tex]\[
m = 2 \quad \text{and} \quad c = 3
\][/tex]
