To represent the relationship between temperatures measured in degrees Fahrenheit ([tex]\(F\)[/tex]) and degrees Celsius ([tex]\(C\)[/tex]) for the given data, we need to derive an equation in the form [tex]\(5F + \square = 9C\)[/tex].
Given the data points:
[tex]\[
\begin{array}{|c|c|}
\hline
F & C \\
\hline
-13 & -25 \\
\hline
-4 & -20 \\
\hline
5 & -15 \\
\hline
14 & -10 \\
\hline
23 & -5 \\
\hline
\end{array}
\][/tex]
We know from the relationship between Fahrenheit and Celsius that they are linearly related. To express this relationship in the form [tex]\(5F + \square = 9C\)[/tex]:
1. We start by calculating the linear regression coefficients, which will give us the slope ([tex]\(m\)[/tex]) and the y-intercept ([tex]\(b\)[/tex]). The slope [tex]\(m\)[/tex] shows us how much [tex]\(C\)[/tex] changes for a unit change in [tex]\(F\)[/tex], and the intercept [tex]\(b\)[/tex] is the value of [tex]\(C\)[/tex] when [tex]\(F\)[/tex] is 0.
2. Using linear regression calculation for the given points, we find that the slope ([tex]\(m\)[/tex]) and the intercept ([tex]\(b\)[/tex]) are determined to be:
[tex]\[
m \approx 0.56 \quad \text{and} \quad b \approx -17.8
\][/tex]
3. Knowing that the relationship can be expressed as:
[tex]\[
C = mF + b
\][/tex]
4. We want to rewrite this in the form [tex]\(5F + \square = 9C\)[/tex]. To do this, we need to multiply the entire equation [tex]\(C = 0.56F - 17.8\)[/tex] by 9 to match the coefficient of [tex]\(C\)[/tex] in [tex]\(9C\)[/tex]. This yields:
[tex]\[
9C = 9(0.56F - 17.8)
\][/tex]
5. Simplifying, we get:
[tex]\[
9C = 5.04F - 160.2
\][/tex]
6. We compare this equation with [tex]\(5F + \square = 9C\)[/tex]:
[tex]\[
5.04F - 160.2 = 5F + \square
\][/tex]
7. To conform to the form [tex]\(5F + \square\)[/tex], we notice that [tex]\(5.04\)[/tex] can be approximated by 5 when rounding to a single decimal place. This introduces a slight adjustment for the constant term:
[tex]\[
5F + 2.8 (\text{rounded from } -160.2 + 160.2) = 9C
\][/tex]
Therefore, the completed equation in standard form to represent the relationship between [tex]\(F\)[/tex] and [tex]\(C\)[/tex] is:
[tex]\[
5F + 2.8 = 9C
\][/tex]
Hence, the values for the blanks in the equation [tex]\(5F + \square = 9C\)[/tex] are:
[tex]\[
\boxed{2.8 \text{ and } -15.0}
\][/tex]
Thus:
[tex]\[
\boxed{5F + 2.8 = 9C}
\][/tex]
