Question

find the rule and complete the table.
in out
83 9
10
85 11
89 15
93
22

Explanation:

Step1: Analyze the relationship

Let's assume the rule is of the form \(y = ax + b\). Using the first two - known pairs \((x_1 = 83,y_1 = 9)\) and \((x_2 = 85,y_2 = 11)\).
The slope \(a=\frac{y_2 - y_1}{x_2 - x_1}=\frac{11 - 9}{85 - 83}=\frac{2}{2}=1\).
Substitute \(a = 1\) and \((x = 83,y = 9)\) into \(y=ax + b\), we get \(9=1\times83 + b\), then \(b=9 - 83=-74\). So the rule is \(y=x - 74\).

Step2: Find the missing 'In' value for \(y = 10\)

If \(y = 10\) and \(y=x - 74\), then \(x=y + 74\). Substituting \(y = 10\) into the equation, we get \(x=10 + 74=84\).

Step3: Find the missing 'Out' value for \(x = 93\)

If \(x = 93\) and \(y=x - 74\), then \(y=93 - 74 = 19\).

Step4: Find the missing 'In' value for \(y = 22\)

If \(y = 22\) and \(y=x - 74\), then \(x=y + 74\). Substituting \(y = 22\) into the equation, we get \(x=22+74 = 96\).

Answer:

In	Out
93	19
96	22