To fill in the missing value of [tex]\( y \)[/tex] when [tex]\( x = 7 \)[/tex], let's carefully analyze the pattern and steps involved:
1. Identify the Changes in [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
- From [tex]\( x = 3 \)[/tex] to [tex]\( x = 6 \)[/tex]:
- Change in [tex]\( x \)[/tex] ([tex]\(\Delta x_1\)[/tex]) = [tex]\(6 - 3 = 3\)[/tex]
- Change in [tex]\( y \)[/tex] ([tex]\(\Delta y_1\)[/tex]) = [tex]\(-17 - 0 = -17\)[/tex]
- From [tex]\( x = 6 \)[/tex] to [tex]\( x = 9 \)[/tex]:
- Change in [tex]\( x \)[/tex] ([tex]\(\Delta x_2\)[/tex]) = [tex]\(9 - 6 = 3\)[/tex]
- Change in [tex]\( y \)[/tex] ([tex]\(\Delta y_2\)[/tex]) = [tex]\(-34 - (-17) = -17\)[/tex]
- From [tex]\( x = 9 \)[/tex] to [tex]\( x = 10 \)[/tex]:
- Change in [tex]\( x \)[/tex] ([tex]\(\Delta x_3\)[/tex]) = [tex]\(10 - 9 = 1\)[/tex]
- Change in [tex]\( y \)[/tex] ([tex]\(\Delta y_3\)[/tex]) = [tex]\(-51 - (-34) = -17\)[/tex]
2. Confirm the Consistent Rate of Change in [tex]\( y \)[/tex]:
- The change in [tex]\( y \)[/tex] for each unit change in [tex]\( x \)[/tex] from the patterns observed is [tex]\(-17\)[/tex].
3. Calculate [tex]\( y \)[/tex] for [tex]\( x = 7 \)[/tex]:
- The change in [tex]\( x \)[/tex] from [tex]\( x = 3 \)[/tex] to [tex]\( x = 7 \)[/tex]:
- [tex]\(\Delta x_{required}\)[/tex] = [tex]\(7 - 3 = 4\)[/tex]
- Change in [tex]\( y \)[/tex] when [tex]\( \Delta x \)[/tex] changes by 4 units:
- Since for each 1 unit increase in [tex]\( x \)[/tex], [tex]\( y \)[/tex] decreases by 5.67 (approximated as [tex]\(\frac{-17}{3} = -5.6667\)[/tex]),
- Total change in [tex]\( y \)[/tex]:
- [tex]\(\Delta y_{required}\)[/tex] = [tex]\((\frac{-17}{3}) \times 4 = -22.666666666666668\)[/tex]
4. Find the missing [tex]\( y \)[/tex] when [tex]\( x = 7 \)[/tex]:
- Starting at [tex]\( y = 0 \)[/tex] when [tex]\( x = 3 \)[/tex],
- [tex]\( y_{required} = 0 + (-22.666666666666668) = -22.666666666666668\)[/tex]
So, the missing value when [tex]\( x = 7 \)[/tex] is approximately [tex]\(-22.67\)[/tex].
In summary, the filled-in table with the missing values is:
[tex]\[
\begin{array}{|c|c|}
\hline x & y \\
\hline 3 & 0 \\
\hline 6 & -17 \\
\hline 9 & -34 \\
\hline 10 & -51 \\
\hline 7 & -22.67 \\
\hline
\end{array}
\][/tex]
