Problem 61 - Elementary and Intermediate Algebra

Question

Solve for the specified variable or expression. $$ S n=(n-2) 180 \text { for } n $$

Step-by-Step Solution

Answer

The solution for \( n \) is \( n = \frac{S_n + 360}{180} \).

Step 1: Isolate the terms with the variable n

The given equation is \( S_n = (n-2) \times 180 \). Begin by expanding and simplifying to make it easier to solve for \( n \). Isolate the term involving \( n \), so write it as \( S_n = 180n - 360 \).

Step 2: Move constants to the other side

Add 360 to both sides of the equation to get rid of the constant on the right. This results in \( S_n + 360 = 180n \).

Step 3: Solve for n

Finally, to solve for \( n \), divide both sides of the equation by 180: \[ n = \frac{S_n + 360}{180}. \] This gives the expression for \( n \) in terms of \( S_n \).

Key Concepts

Equation Solving

Equation solving is like unlocking a mystery. It's the process of finding the value of a variable that makes a mathematical sentence true. In algebra, this often means getting the variable on one side of the equation by itself. For example, when faced with an equation like \( S_n = (n-2) \times 180 \), the goal is to figure out what \( n \) is. To solve it:

• Start by identifying the equation you need to work with.
• Next, perform operations that simplify the equation step by step.
• Finally, isolate the variable and solve for it.

This process can include operations like addition, subtraction, multiplication, or division, as needed, to simplify and isolate the variable.

Variables in Algebra

Variables are like placeholders in algebra. They stand for unknown values and are usually represented by letters such as \( n, x, y \) or any other letter. A major part of understanding algebra is knowing how these variables interact with numbers and operations.In the equation \( S_n = (n-2) \times 180 \):

• \( S_n \) is a placeholder for a specific number that depends on \( n \).
• The \( n \) in the equation represents the number we are trying to solve.

The beauty of using variables in algebra is that it allows us to work with complex problems. We can express relationships and changes in a way that is simple, yet powerful. Algebra allows us to tackle questions about relationships and patterns that would otherwise be difficult to solve.

Isolation of Variables

The isolation of variables is a key technique in algebra. It involves rearranging an equation to get a variable by itself on one side of the equation, making it simpler to find its value. This technique is crucial for solving equations efficiently.In our example, \( S_n = 180n - 360 \):

• The term \( 180n \) is separated from \( -360 \) by adding 360 to both sides, keeping the equation balanced.
• Once constants are moved, we isolate \( n \) by dividing the whole equation by 180.

The result is that \( n \) is expressed in terms of \( S_n \), giving \( n = \frac{S_n + 360}{180} \). Isolation of variables is powerful because it reduces complex equations into simple expressions that are much easier to work with.