Question

Solve each equation. \(\frac{6}{x}-\frac{1}{10}=\frac{1}{5}\)

Short answer

x = 20

Step-by-step solution

- Identify the common denominators

To solve the equation \(\frac{6}{x}-\frac{1}{10}=\frac{1}{5}\), first identify the common denominators of the fractions involved. The common denominators for the fractions are the least common multiple (LCM) of the numbers involved. In this case, the denominators are \(x\), \(10\), and \(5\).

- Clear the fractions

Multiply each term in the equation by the LCM of the denominators, which is \(10x\): \(\frac{6}{x} \times 10x - \frac{1}{10} \times 10x = \frac{1}{5} \times 10x\). This simplifies to \(60 - x = 2x\).

- Solve for x

Isolate \(x\) by adding \(x\) to both sides: \(60 - x + x = 2x + x\). Simplify to obtain \(60 = 3x\).

- Finalize the solution

Solve for \(x\) by dividing both sides by 3: \(x = \frac{60}{3} = 20\). Thus, \(x = 20\).

- Verify the solution

Substitute \(x = 20\) back into the original equation to verify: \(\frac{6}{20} - \frac{1}{10} = \frac{1}{5}\). Simplifying both sides confirms that the left-hand side equals the right-hand side, verifying our solution.

Key concepts

least common multiple

When solving equations involving fractions, it's useful to find the least common multiple (LCM) of the denominators. The LCM is the smallest number that each denominator can divide without leaving a remainder. In our example, the denominators are \(x\), \(10\), and \(5\). To find their LCM, consider the following steps:

• List the multiples of each number.
• Identify the smallest common multiple among those lists.

For instance, the multiples of 10 are 10, 20, 30, etc., and the multiples of 5 are 5, 10, 15, etc. Since \(x\) is also a denominator, we need to include some value of \(x\) that will fit in this pattern. Thus, one way is to assume \(10x\) as the LCM for simplicity, making computation easier. This way, we can correctly combine and eliminate the fractions in the equation efficiently.

isolating variables

Isolating the variable is a crucial part of solving equations. It means getting the unknown (variable) alone on one side of the equation to determine its value. In our example, after clearing the fractions, the equation becomes:
\(60 - x = 2x\).
To isolate \(x\), follow these steps:

• Add \(x\) to both sides: \(60 - x + x = 2x + x\).
• Simplify to get: \(60 = 3x\).

Once simplified, divide both sides by 3 to get \(x\). This step reveals that \(x = 20\).
By isolating \(x\), we successfully determine its value.

clearing fractions

Clearing fractions from an equation simplifies it, making it easier to solve. The process involves eliminating the denominators by multiplying through by the LCM. In our equation, the denominators are \(x\), \(10\), and \(5\). After finding the LCM as \(10x\), follow these steps:

• Multiply each term by the LCM: \( \frac{6}{x} \times 10x - \frac{1}{10} \times 10x = \frac{1}{5} \times 10x \).
• Simplify each term: \(60 - x = 2x\).

By clearing the fractions, we quickly reduce the equation to a simpler form and proceed to isolate the variable.

verifying solutions

Verifying solutions ensures the calculated value satisfies the original equation. To verify our solution, substitute \(x = 20\) back into the initial equation:
\(\frac{6}{20} - \frac{1}{10} = \frac{1}{5}\).
Simplify the left side:

• \(\frac{6}{20} = \frac{3}{10}\)
• \(\frac{3}{10} - \frac{1}{10} = \frac{2}{10} = \frac{1}{5}\)

Both sides yield \(\frac{1}{5}\), confirming the solution is correct. Verification checks if the assumption and isolation steps lead to a legitimate and accurate result.