Problem 17 - Elementary Algebra

Question

\(\frac{5}{3 n}-\frac{1}{9}=\frac{1}{n}\)

Step-by-Step Solution

Answer

The value of \(n\) is 6.

Step 1: Clear the Denominators

First, identify the common denominator for the fractions, which is \(9n\). Multiply each term of the equation by \(9n\) to eliminate the fractions: \(9n \times \frac{5}{3n} - 9n \times \frac{1}{9} = 9n \times \frac{1}{n}\). This simplifies to: \(3 imes 5 - n = 9\). So, the equation becomes \(15 - n = 9\).

Step 2: Isolate the Variable

To solve for \(n\), add \(n\) to both sides of the equation to move it to the right side:\(15 = n + 9\).

Step 3: Solve the Equation

Subtract \(9\) from both sides to isolate \(n\): \(15 - 9 = n\). Therefore, \(n = 6\).

Step 4: Verify the Solution

Substitute \(n = 6\) back into the original equation to check if it satisfies the equation:\(\frac{5}{3 \times 6}-\frac{1}{9} = \frac{1}{6}\). This simplifies to \(\frac{5}{18} - \frac{2}{18} = \frac{3}{18}\), which is the same as \(\frac{1}{6}\), confirming the solution is correct.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in elementary algebra. It's like solving a puzzle where each piece must fit to reveal the correct solution.
In equations, we have variables that represent unknown values, and the goal is to determine their value. This often involves simplifying both sides of the equation and performing operations that maintain the balance of the equation.
A key principle is that whatever operation you perform on one side, you must also perform on the other side.

• This can include:
  • Adding or subtracting the same number from both sides
  • Multiplying or dividing both sides by the same number (except zero)

Always remember, equations represent a balance, like a scale. Our job in solving an equation is to keep that scale perfectly balanced while we figure out the missing value.

Fractions

Fractions can seem tricky at first, but they are just another way to represent numbers.
When dealing with fractions in equations, it's important to remember that a fraction is just a division problem: the numerator is divided by the denominator.
One way to simplify equations with fractions is to eliminate the fractions altogether.
This is usually done by finding a common denominator, which allows the fractions to be rewritten with the same bottom number, making them easier to combine or eliminate.
In our exercise example, we used a similar method to clear the fractions by multiplying the entire equation by the least common multiple of the denominators. This step helps to convert the equation into an easier form without fractions, making it simpler to solve.
Always simplify the fractions if possible before further calculations, either by reducing them or by applying operations such as multiplication or division.

Variable Isolation

Variable isolation is the technique of moving all terms containing the variable to one side of the equation.
Simply put, it's like trying to get the variable 'alone' on one side.
This is crucial because it allows us to find the value of the variable without confusion.
In practice, we achieve this by using inverse operations. For example:

• If a term with a variable is being added, subtract it from both sides.
• If it's being subtracted, add it to both sides.
• If it's being multiplied, divide both sides by that number.
• And if it's divided, multiply both sides accordingly.

In our equation, we moved the term containing the variable to one side by simply adding or subtracting as needed.
Once isolated, it's easy to see what operation will solve for the variable.

Verification of Solution

Verification is an essential final step in solving algebraic equations. It acts like a safety net to ensure the solution is correct.
To verify a solution, substitute the value back into the original equation to see if it creates a true statement.
This step confirms that the solution works not just in our modified version of the equation, but in the original setup as well.

• Replace the variable in the equation with the value you found.
• Simplify both sides to verify they are equal.

In the given example, after finding that the variable \( n = 6 \), we substituted back into the original equation and found that both sides equate, confirming the correctness of our solution. It's important to always perform this check. It guarantees the reliability of our solution and highlights any possible errors in calculations.