One carpet cleaning cost are shown in the table. the relationship is linear. find and interpret the rate of change and the initial value for this situation.

I see from the given data that each additional carpet after the first costs $50 each to have cleaned.

The first costs $125.  Supposing we subtract $50 from that, we get $75.  So, it sounds like it costs $75 just to get the carpet cleaners to your house.

The rule would then be $75 + $50x, where x : {1, 2, 3, 4, }

Let's check this result.  Suppose x = 1.  Will the charge be $125, as show in the table?  $75 + $50(1) = $125.  Correct.

Suppose x = 4.  Will the charge be $275 as shown in the table?
   $75 + $50(4) = $275.  YES