To determine the relationship between the given x and y values and to predict the missing y value for \(x = 5\), we start by assuming that y follows a linear relationship with x. This can be represented by the equation of a line:
[tex]\[ y = mx + c \][/tex]
where \(m\) is the slope of the line and \(c\) is the y-intercept.
### Step-by-Step Solution:
1. Identify the given values:
We have the following data points:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 & 5 \\
\hline
y & 13 & 21 & 29 & 37 & ? \\
\hline
\end{array}
\][/tex]
2. Formulate the system of equations:
From the given points (1, 13), (2, 21), (3, 29), and (4, 37), we set up the following system of equations based on the line equation \( y = mx + c \):
[tex]\[
\begin{cases}
13 = m \cdot 1 + c \\
21 = m \cdot 2 + c \\
29 = m \cdot 3 + c \\
37 = m \cdot 4 + c
\end{cases}
\][/tex]
3. Find the slope (m) and y-intercept (c):
By solving these equations, we get the following results for the coefficients:
[tex]\[
m = 8, \quad c = 5
\][/tex]
Substituting these values back into the line equation, we get:
[tex]\[
y = 8x + 5
\][/tex]
4. Predict the missing y value for \(x = 5\):
We now use the derived equation of the line to predict the y value when \(x = 5\):
[tex]\[
y = 8 \cdot 5 + 5
\][/tex]
[tex]\[
y = 40 + 5
\][/tex]
[tex]\[
y = 45
\][/tex]
### Final Answer:
The coefficients of the line are:
[tex]\[ m = 8, \quad c = 5 \][/tex]
The predicted y value for \(x = 5\) is:
[tex]\[
y = 45
\][/tex]
Thus, the completed table is:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & 1 & 2 & 3 & 4 & 5 \\
\hline
y & 13 & 21 & 29 & 37 & 45 \\
\hline
\end{array}
\][/tex]
