Problem 22 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the given equation. $$ \frac{1}{x}+\frac{2}{x}=6 $$

Step-by-Step Solution

Answer

The solution to the given equation \(\frac{1}{x} + \frac{2}{x} = 6\) is \(x = \frac{1}{2}\).

Step 1: Combine the fractions on the left side

Add the two fractions on the left side of the equation with the same denominator. The given equation is: \[ \frac{1}{x} + \frac{2}{x} = 6 \] To add the fractions, we will keep the common denominator and add the numerators: \[ \frac{1+2}{x} = 6 \]

Step 2: Simplify the fraction

After adding the numerators, we will simplify the fraction: \[ \frac{3}{x} = 6 \]

Step 3: Eliminate the fraction by multiplying both sides by the common denominator

To eliminate the fraction, we will multiply both sides of the equation by x. This results in: \[ x \cdot \frac{3}{x} = 6 \cdot x \] The x in the numerator and denominator will cancel, and the equation becomes: \[ 3 = 6x \]

Step 4: Solve for x

Finally, solve for x by dividing both sides of the equation by 6: \[ \frac{3}{6} = x \] Simplify the fraction: \[ x = \frac{1}{2} \] So, \(x = \frac{1}{2}\) is the solution to the given equation.

Key Concepts

Fraction Simplification

Simplifying fractions is a fundamental step in solving algebraic equations that involve fractions. When both fractions have the same denominator, adding them is straightforward. Keep the common denominator and add the numerators directly.
In our case, we started with the equation:

• \(\frac{1}{x} + \frac{2}{x} = 6\)

The fractions involved, \(\frac{1}{x}\) and \(\frac{2}{x}\), share the same denominator \(x\). Adding the numerators \(1 + 2\) gives us:

• \(\frac{3}{x}\)

This process is called fraction simplification by addition. Simplifying fractions helps in clearing the complexity of the expression, paving the way for easier algebraic manipulation.

Equation Solving Steps

The process of solving equations involves a series of logical steps aimed at isolating the unknown variable. Consider our original equation after simplification:

• \(\frac{3}{x} = 6\)

To remove the fraction and solve for \(x\), multiply both sides by the denominator, \(x\). This action effectively cancels out \(x\) from the fraction:

• \(x \cdot \frac{3}{x} = 6 \cdot x\)
• Resulting in: \(3 = 6x\)

Next, isolate \(x\) by performing inverse operations. Divide both sides by 6:

• \(\frac{3}{6} = x\)
• Thus, \(x = \frac{1}{2}\)

This sequence of steps—simplifying, eliminating fractions, and isolating the variable—ensures a methodical approach to solving algebraic equations.

Common Denominators

Understanding and using common denominators is crucial when adding or subtracting fractions. A common denominator is simply a shared denominator between two or more fractions, enabling seamless addition or subtraction.
In our exercise, both fractions had the same denominator \(x\). This allowed us to combine them easily:

• \(\frac{1}{x} + \frac{2}{x} = \frac{3}{x}\)

However, if fractions had different denominators, finding a common denominator would be necessary. This typically involves identifying the least common multiple (LCM) of the denominators and adjusting the fractions accordingly.
Doing so standardizes the denominators, allowing us to sum or subtract the fractions just like whole numbers. Understanding this concept is vital for clarity in operations involving fractions.