Okay, so I need to simplify the expression 19x + 1 - 4x - 81. Let me think about how to approach this step by step. First, I remember that simplifying an algebraic expression involves combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, the terms with x are 19x and -4x, and the constant terms are 1 and -81.

Let me start by identifying the like terms. The terms with x are 19x and -4x. The constants are 1 and -81. So I can group these together. For the x terms, I can combine 19x and -4x. To do that, I just add their coefficients. 19 minus 4 is 15, so that would be 15x. Then, for the constants, 1 minus 81 is... let me calculate that. 1 minus 81 is -80. So putting those together, the simplified expression should be 15x - 80.

Wait, let me double-check that. Combining 19x and -4x gives 15x, that's correct. Then combining 1 and -81, which is 1 - 81, which is indeed -80. So yes, the simplified form is 15x - 80. Hmm, that seems straightforward. Is there any chance I made a mistake here?

Let me verify again. Starting with the original expression: 19x + 1 - 4x - 81. Let's rearrange the terms to group the like terms together. So, 19x - 4x + 1 - 81. Calculating the coefficients: 19 - 4 is 15, so 15x. Then 1 - 81 is -80. So yes, that's the same result. Okay, that checks out.

Another way to look at it is to distribute the signs if there are any parentheses, but in this case, there aren't any. So it's just combining the coefficients directly. Let me also think if there's any property of algebra that I might be overlooking here. The associative and commutative properties allow us to rearrange and group terms as needed. Since addition is commutative, we can rearrange the terms in any order. So 19x + 1 - 4x - 81 is the same as 19x - 4x + 1 - 81. Then combining like terms as I did before.

Alternatively, if I factor out an x from the terms with x, but since there are only two such terms, that might not be necessary. But in this case, factoring might complicate things more. Let me see: 19x - 4x is 15x, so factoring x would give x(15) + (1 - 81). Which is the same as 15x - 80. So that's another way to look at it, but it's essentially the same process.

I think that's thorough. No steps seem missing. The key is to combine the coefficients of x and the constants separately. So the simplified expression is 15x - 80. I don't see any errors here. The arithmetic checks out: 19 minus 4 is 15, and 1 minus 81 is -80. Yep, that's right. So the final answer should be 15x minus 80.
Therefore, the simplified expression is $\boxed{15x - 80}$.
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