<Question>
Simplify the following expression in $x$: $$19x + 1 - 4x - 81.$$
</Question>

<think>
<Plan_1>
Okay, so I need to simplify the expression 19x + 1 - 4x - 81. Let me think about how to approach this. Alright, simplifying algebraic expressions usually involves combining like terms. Like terms are terms that have the same variable raised to the same power. So, in this case, the terms with x should be combined together, and the constant numbers should be combined together.

First, let me identify the terms in the expression. The expression has four terms: 19x, 1, -4x, and -81. So, the terms with x are 19x and -4x. The constant terms are 1 and -81.

Let me start with the x terms. Combining 19x and -4x.
Since they are like terms, I can add their coefficients. The coefficient of the first term is 19, and the second term is -4. So, 19 + (-4) equals... let's see, 19 minus 4 is 15. So, combining those gives 15x.

Now, moving on to the constant terms: 1 and -81. Again, these are like terms because they are both constants without any variables.
Adding those together: 1 + (-81) is the same as 1 - 81, which equals -80.

Now, putting those two results together, we have 15x from the x terms and -80 from the constants.
So, combining them gives 15x - 80.

Let me double-check to make sure I didn't make any mistakes. For the x terms: 19x minus 4x.

<prompt_1.1> Re-evaluate the combination of x terms and constants to verify the simplification. </prompt_1.1>

Wait, just to be thorough, let me write out the original expression and then group the like terms step by step.

<prompt_1.2> Write out the original expression and group the like terms step by step. </prompt_1.2>

Another way to check is to substitute a value for x in the original expression and in the simplified expression to see if they yield the same result. Let's pick x = 1.

<prompt_1.3> Substitute x = 1 and x = 0 into the original and simplified expressions to verify the simplification. </prompt_1.3>
</Plan_1>

<Execution_1>
<execution_1.1>
19 minus 4 is 15, so that's correct. For the constants: 1 minus 81 is indeed -80. So, combining them gives 15x - 80.
</execution_1.1>

<execution_1.2>
Original expression: 19x + 1 - 4x - 81.

Group the x terms and the constant terms:
(19x - 4x) + (1 - 81).

Calculating each group:
19x - 4x = 15x.

1 - 81 = -80.

So, combining them gives 15x - 80.
</execution_1.2>

<execution_1.3>
Original: 19(1) + 1 - 4(1) - 81 = 19 + 1 - 4 - 81 = 20 - 4 = 16; 16 - 81 = -65.

Simplified: 15(1) - 80 = 15 - 80 = -65. Okay, that matches. Let's try x = 0.

Original: 19(0) + 1 - 4(0) - 81 = 0 + 1 - 0 - 81 = 1 - 81 = -80.

Simplified: 15(0) - 80 = 0 - 80 = -80. That also matches.
</execution_1.3>
</Execution_1>

<Final_answer>
Therefore, the simplified expression is $\boxed{15x-80}$.
</Final_answer>
</think>