Problem 13 - Elementary Algebra

Question

Solve. $$ 13=5 n $$

Step-by-Step Solution

Answer

\( n = \frac{13}{5} \)

Step 1: Identify the Equation

The equation given is \( 13 = 5n \). Our goal is to find the value of \( n \).

Step 2: Isolate the Variable

To isolate \( n \), divide both sides of the equation by 5. This gives us \( n = \frac{13}{5} \).

Step 3: Simplify the Solution

The fraction \( \frac{13}{5} \) is already in its simplest form, so \( n = \frac{13}{5} \) is the final solution.

Key Concepts

Variable Isolation

When solving linear equations, one of the first steps is variable isolation. This means getting the variable alone on one side of the equation. In the problem provided, the equation is given as \( 13 = 5n \).
To isolate the variable \( n \), we need to remove any coefficients or constants that are hindering \( n \) from being by itself. Here, \( n \) is multiplied by 5. So, we should do the opposite operation to both sides of the equation. This is called the inverse operation, which, in this case, is division.

To isolate \( n \), divide both sides of the equation by 5. This operation cancels out the 5 on the right side of the equation, leaving \( n \) by itself:
\[ n = \frac{13}{5} \]
By performing this simple division, we have successfully isolated \( n \). This step is essential because it leads you straight to the solution of the equation.

Simplifying Fractions

Once you've isolated the variable and obtained it in terms of a fraction, the next step is to simplify the fraction, if possible. This involves reducing the fraction to its simplest form. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1.

Looking at our solution where \( n = \frac{13}{5} \), we notice that 13 is a prime number and does not share any common factors with 5. Therefore, \( \frac{13}{5} \) is already as simple as it can get.
Some students might wonder why we need to simplify fractions at all. Here are a few reasons:

• Simplified fractions are often easier to interpret.
• They provide a cleaner and more elegant result.
• It's generally considered mathematically neat and correct.

So, remember to always check if you can simplify a fraction after isolating the variable for a final polished answer.

Equation Manipulation

Equation manipulation is a necessary skill in solving linear equations. It's about applying mathematical operations to both sides of an equation to make it easier to interpret or solve.
In our original exercise, we started with the equation \( 13 = 5n \). Our goal was to manipulate this equation to solve for \( n \).

Here's a step-by-step breakdown of the manipulation process:

• Identify the operation that\( n \) is involved in, which here is multiplication by 5.
• Determine the inverse operation, which is division, and apply it to both sides of the equation to maintain balance within the equation.
• After the division, we get \( n = \frac{13}{5} \), where the variable is isolated, and the equation is manipulated into a simpler form.

This technique is not only applicable in this problem but also a foundational element of solving any equation in algebra. By consistently practicing equation manipulation, students become adept at identifying the necessary steps to simplify and solve equations correctly.