To determine the value of [tex]\( n \)[/tex] for which [tex]\( f(n) = 20 \)[/tex] given the function [tex]\( f(x) \)[/tex], we first need to find the equation of the linear function. Here is the systematic approach to solve this problem:
1. Identify the given points:
We are given two points on the line:
[tex]\[ (-4, -25) \][/tex]
[tex]\[ (-1, -10) \][/tex]
2. Calculate the slope (m) of the linear function. The slope [tex]\( m \)[/tex] can be found using the formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Plugging in the given points, we get:
[tex]\[
m = \frac{-10 - (-25)}{-1 - (-4)} = \frac{-10 + 25}{-1 + 4} = \frac{15}{3} = 5
\][/tex]
3. Formulate the equation of the line using the point-slope form, which states [tex]\( y - y_1 = m(x - x_1) \)[/tex]. Using the slope [tex]\( m = 5 \)[/tex] and one of the points [tex]\((x_1, y_1) = (-4, -25)\)[/tex]:
[tex]\[
y - (-25) = 5(x - (-4))
\][/tex]
Simplifying this, we get:
[tex]\[
y + 25 = 5(x + 4)
\][/tex]
[tex]\[
y + 25 = 5x + 20
\][/tex]
[tex]\[
y = 5x + 20 - 25
\][/tex]
[tex]\[
y = 5x - 5
\][/tex]
4. Substitute [tex]\( y = 20 \)[/tex] into the equation [tex]\( y = 5x - 5 \)[/tex] to solve for [tex]\( x \)[/tex]:
[tex]\[
20 = 5x - 5
\][/tex]
[tex]\[
20 + 5 = 5x
\][/tex]
[tex]\[
25 = 5x
\][/tex]
[tex]\[
x = \frac{25}{5} = 5
\][/tex]
Therefore, the value of [tex]\( n \)[/tex] is [tex]\(\boxed{5}\)[/tex].
