Answer :
We want to find a linear function of the form
[tex]$$
y = m x + b,
$$[/tex]
where [tex]$m$[/tex] is the slope and [tex]$b$[/tex] is the [tex]$y$[/tex]-intercept. The data given is:
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
0 & 100 \\
1 & 86 \\
2 & 65 \\
3 & 59 \\
4 & 41 \\
5 & 34 \\
\end{array}
\][/tex]
Step 1. Compute the slope [tex]$m$[/tex].
The formula for the slope is
[tex]$$
m = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2},
$$[/tex]
where [tex]$\bar{x}$[/tex] and [tex]$\bar{y}$[/tex] are the averages of the [tex]$x$[/tex] and [tex]$y$[/tex] values respectively.
After performing the calculations, the slope is found to be approximately
[tex]$$
m \approx -13.46.
$$[/tex]
Step 2. Compute the [tex]$y$[/tex]-intercept [tex]$b$[/tex].
Once the slope is determined, the [tex]$y$[/tex]-intercept can be calculated using the formula
[tex]$$
b = \bar{y} - m\bar{x}.
$$[/tex]
With the computed values, the intercept is approximately
[tex]$$
b \approx 97.81.
$$[/tex]
Step 3. Formulate the linear equation.
Substituting the values of [tex]$m$[/tex] and [tex]$b$[/tex] into the linear equation gives
[tex]$$
y \approx -13.46\,x + 97.81.
$$[/tex]
Rounding the slope to [tex]$-13.5$[/tex] and the intercept to [tex]$97.8$[/tex], the equation becomes
[tex]$$
y = -13.5\,x + 97.8.
$$[/tex]
Step 4. Identify the correct model.
Comparing with the answer choices:
[tex]\[
\begin{aligned}
\textbf{A: } & y=-13.5 x+97.8 \\
\textbf{B: } & y=-13.5 x+7.3 \\
\textbf{C: } & y=97.8 x-13.5 \\
\textbf{D: } & y=7.3 x-97.8
\end{aligned}
\][/tex]
we see that the equation we obtained, [tex]$y=-13.5 x+97.8$[/tex], corresponds to option A.
Final Answer: Option A.
[tex]$$
y = m x + b,
$$[/tex]
where [tex]$m$[/tex] is the slope and [tex]$b$[/tex] is the [tex]$y$[/tex]-intercept. The data given is:
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
0 & 100 \\
1 & 86 \\
2 & 65 \\
3 & 59 \\
4 & 41 \\
5 & 34 \\
\end{array}
\][/tex]
Step 1. Compute the slope [tex]$m$[/tex].
The formula for the slope is
[tex]$$
m = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2},
$$[/tex]
where [tex]$\bar{x}$[/tex] and [tex]$\bar{y}$[/tex] are the averages of the [tex]$x$[/tex] and [tex]$y$[/tex] values respectively.
After performing the calculations, the slope is found to be approximately
[tex]$$
m \approx -13.46.
$$[/tex]
Step 2. Compute the [tex]$y$[/tex]-intercept [tex]$b$[/tex].
Once the slope is determined, the [tex]$y$[/tex]-intercept can be calculated using the formula
[tex]$$
b = \bar{y} - m\bar{x}.
$$[/tex]
With the computed values, the intercept is approximately
[tex]$$
b \approx 97.81.
$$[/tex]
Step 3. Formulate the linear equation.
Substituting the values of [tex]$m$[/tex] and [tex]$b$[/tex] into the linear equation gives
[tex]$$
y \approx -13.46\,x + 97.81.
$$[/tex]
Rounding the slope to [tex]$-13.5$[/tex] and the intercept to [tex]$97.8$[/tex], the equation becomes
[tex]$$
y = -13.5\,x + 97.8.
$$[/tex]
Step 4. Identify the correct model.
Comparing with the answer choices:
[tex]\[
\begin{aligned}
\textbf{A: } & y=-13.5 x+97.8 \\
\textbf{B: } & y=-13.5 x+7.3 \\
\textbf{C: } & y=97.8 x-13.5 \\
\textbf{D: } & y=7.3 x-97.8
\end{aligned}
\][/tex]
we see that the equation we obtained, [tex]$y=-13.5 x+97.8$[/tex], corresponds to option A.
Final Answer: Option A.