Question

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

Explanation:

Step1: Analyze the given pairs

Let's assume the rule is of the form \(y = ax + b\). Using the pairs \((83,9)\) and \((85,11)\). First, find the slope \(a\). The slope formula is \(a=\frac{y_2 - y_1}{x_2 - x_1}\). Here, \(x_1 = 83,y_1 = 9,x_2=85,y_2 = 11\). So \(a=\frac{11 - 9}{85 - 83}=\frac{2}{2}=1\).

Step2: Find the value of \(b\)

Substitute \(a = 1\) and the point \((83,9)\) into \(y=ax + b\), we get \(9=1\times83 + b\). Solving for \(b\): \(b=9 - 83=- 74\). So the rule is \(y=x - 74\).

Step3: Find the missing 'In' value for 'Out' = 10

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

Step4: Find the missing 'Out' value for 'In' = 93

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

Step5: Find the missing 'In' value for 'Out' = 22

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

Answer:

In	Out
93	19
96	22