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The first step in solving an algebra word problem is to identify and define the variables you will use. For the hotel room problem, we need to find the number of smoking and nonsmoking rooms. To do this, we define a variable to represent the unknown quantity, which is often given in the problem statement itself. In this case, let’s define the variable as follows: let the number of smoking rooms be denoted by \( s \). According to the problem, the number of nonsmoking rooms is seven times the number of smoking rooms. Therefore, we can denote the number of nonsmoking rooms as \( 7s \). This step is crucial because it translates the words of the problem into mathematical expressions we can manipulate.

Once we have defined the variables, the next step is to set up an equation based on the relationships described in the problem. We know the total number of rooms in the hotel is 520. We can express this relationship as an equation involving our variables. Since the total number of rooms is the sum of smoking and nonsmoking rooms, we get:
\[ s + 7s = 520 \] This equation mirrors the information given in the problem and provides a foundation for solving for the unknown variable. Setting up equations correctly is vital for accurately solving the problem.

In algebra, it's useful to combine like terms to simplify the equation and make it easier to solve. Like terms are terms that have the same variables and exponents. In our equation, \[ s + 7s = 520 \], the terms \( s \) and \( 7s \) are like terms because they both involve the variable \( s \). We can combine them by adding their coefficients:
\[ 8s = 520 \] This simplifies our equation to involve just one term with \( s \). Combining like terms reduces complexity and helps in solving the equation more straightforwardly.

The final steps involve solving the simplified, linear equation for the variable. From the combined equation \[ 8s = 520 \], we isolate \( s \) by performing inverse operations. Here we divide both sides of the equation by 8:
\[ s = \frac{520}{8} \] By performing the division, we find: \[ s = 65 \] So, there are 65 smoking rooms. To find the number of nonsmoking rooms, we use the relationship established earlier: \[ 7s = 7 \times 65 = 455 \] Thus, the hotel has 455 nonsmoking rooms. Solving linear equations involves performing operations to isolate the variable, enabling us to find its value and subsequently solve related quantities.