To find the relationship between the quantities in the table, we need to examine how the [tex]\( y \)[/tex] values change as the [tex]\( x \)[/tex] values increase. Here's the given table for clarity:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline x & 1 & 2 & 3 & 4 & 5 \\
\hline y & 20 & 40 & A & 80 & B \\
\hline
\end{array}
\][/tex]
First, let's observe the pattern in the [tex]\( y \)[/tex] values:
- For [tex]\( x = 1 \)[/tex], [tex]\( y = 20 \)[/tex]
- For [tex]\( x = 2 \)[/tex], [tex]\( y = 40 \)[/tex]
From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex], [tex]\( y \)[/tex] increases from 20 to 40. This is an increase of 20. It suggests a consistent increment.
Using the observed incremental pattern, we can predict the missing values:
1. To find [tex]\( A \)[/tex] at [tex]\( x = 3 \)[/tex]:
Since the increment when moving from [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex] is 20 (i.e., 40 - 20), we apply the same increment to the next step:
[tex]\[
A = 40 + 20 = 60
\][/tex]
2. To find [tex]\( B \)[/tex] at [tex]\( x = 5 \)[/tex]:
Observing the pattern from [tex]\( x = 4 \)[/tex]:
Since the increment when moving from [tex]\( x = 2 \)[/tex] to [tex]\( x = 4 \)[/tex] is 40 (i.e., 80 - 40), we apply this consistent increment further:
[tex]\[
B = 80 + 40 = 120
\][/tex]
Thus, the relationship can be summarized with a linear function where [tex]\( y \)[/tex] increases by a fixed increment as [tex]\( x \)[/tex] increases. Hence:
[tex]\(
x \to x \times 20 = y
\)[/tex]
Therefore, the missing values in the table:
[tex]\[ A = 60 \][/tex]
[tex]\[ B = 120 \][/tex]
These values match the consistent pattern of [tex]\( y \)[/tex] increasing by 20 for each increment of 1 in [tex]\( x \)[/tex].
