Answer
See below:
Work Step by Step
The two points on the line for the slope calculation are $\left( 5000,65 \right)$ and $\left( 25000,95 \right)$.
The standard line equation in slope-intercept form is as follows
$y=mx+c$
Find the slope of the line passing through the given points using the formula given below:
$\begin{align}
& m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \\
& =\frac{95-65}{25000-5000} \\
& =\frac{30}{20000} \\
& =\frac{3}{2000}
\end{align}$
Put the value of $m$ in the equation of the line in slope-intercept form, so now the equation of the line is
$y=\frac{3}{2000}x+c$
The above line passes through the point $\left( 5000,65 \right)$, so put this point in the above equation to get the value of c
$\begin{align}
& y=\frac{3}{2000}x+c \\
& 65=\frac{3}{2000}\times 5000+c \\
& 65-\frac{15}{2}=c \\
& c=\frac{115}{2}
\end{align}$
Now, put the value of $c$ in the equation of the line so now the equation of the line is
$y=\frac{3}{2000}x+\frac{115}{2}$
Hence, the function $H\left( x \right)$ represents the percentage of people who call themselves happy and x is the per capita income. So, the function is
$H\left( x \right)=\frac{3}{2000}x+\frac{115}{2}$.
