Question

The table shows a school district's enrollment

for two successive years. Write a linear function
using the data (with x representing the year
number), and then use the model to predict the
enrollment in Year 4.
Year 1 - 8295
Year2 -8072

Answer 1

to get the equation of any straight line, we simply need two points off of it, let's use those two in the table

`\begin{array}cc \cline{1-2} \stackrel{Year}{x}&\stackrel{students}{y}\\ \cline{1-2} 1&8295\\ 2&8072\\ \cline{1-2} \end{array}\hspace{5em} (\stackrel{x_1}{1}~,~\stackrel{y_1}{8295})\qquad (\stackrel{x_2}{2}~,~\stackrel{y_2}{8072}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{8072}-\stackrel{y1}{8295}}}{\underset{run} {\underset{x_2}{2}-\underset{x_1}{1}}} \implies \cfrac{ -223 }{ 1 } \implies - 223`

`\begin{array}ll \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{8295}=\stackrel{m}{- 223}(x-\stackrel{x_1}{1}) \\\\\\ y-8295=-223x+223\implies {\Large \begin{array}{llll} y=-223x+8518 \end{array}} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{for year 4, x = 4}}{y=-223(4)+8518} \implies y=7626~students`