Problem 9 - Elementary and Intermediate Algebra

Question

Complete the solution. $$ \begin{aligned} \frac{2}{5}+\frac{7}{3 x} &=\frac{2}{5} \cdot \frac{\underline{\phantom{xx}}}{3 x}+\frac{7}{3 x} \cdot \frac{\underline{\phantom{xx}}}{5} \\ &=\frac{6 x}{\underline{\phantom{xx}}}+\frac{35}{\underline{\phantom{xx}}} \\ &=\frac{6 x+}{15 x} \end{aligned} $$

Step-by-Step Solution

Answer

\(\frac{6x + 35}{15x}\)

Step 1: Find Common Denominator

To combine the fractions \( \frac{2}{5} \) and \( \frac{7}{3x} \) into a single fraction, we need a common denominator. The common denominator of 5 and \(3x\) is \(15x\).

Step 2: Express \(\frac{2}{5}\) with Denominator \(15x\)

Multiply the numerator and the denominator of \(\frac{2}{5}\) by \(3x\): \[\frac{2}{5} = \frac{2 \cdot 3x}{5 \cdot 3x} = \frac{6x}{15x}\]

Step 3: Express \(\frac{7}{3x}\) with Denominator \(15x\)

Multiply the numerator and the denominator of \(\frac{7}{3x}\) by 5: \[\frac{7}{3x} = \frac{7 \cdot 5}{3x \cdot 5} = \frac{35}{15x}\]

Step 4: Addition of Fractions

Now that both fractions have the same denominator, add them: \[\frac{6x}{15x} + \frac{35}{15x} = \frac{6x + 35}{15x}\]

Step 5: Final Expression

The combined expression is \(\frac{6x + 35}{15x}\). This is the final expression of adding the two fractions.

Key Concepts

Common Denominator

When adding fractions, it's essential to have a common denominator, which is the same bottom number in both fractions. This shared denominator ensures that the fractions can be easily combined.
In our problem, we have two fractions: \( \frac{2}{5} \) and \( \frac{7}{3x} \). The denominators here are 5 and \( 3x \).
To find a common denominator:

• Multiply the two denominators together: \( 5 \times 3x = 15x \).

The common denominator for these fractions is \( 15x \). This allows us to rewrite both fractions so that the denominators match. Once this is achieved, we can add the fractions together without any hassle.

Numerator Adjustment

After determining the common denominator for adding fractions, the next step is adjusting the numerators accordingly. This ensures that each fraction accurately represents the same quantity when compared to the new common denominator.

• For the fraction \( \frac{2}{5} \), multiply both the numerator and the denominator by \( 3x \). This gives: \( \frac{2 \cdot 3x}{5 \cdot 3x} = \frac{6x}{15x} \).
• For \( \frac{7}{3x} \), multiply both the numerator and the denominator by 5. That results in: \( \frac{7 \cdot 5}{3x \cdot 5} = \frac{35}{15x} \).

By making these adjustments, we've changed the fractions to new forms \( \frac{6x}{15x} \) and \( \frac{35}{15x} \) which are now ready to be combined into a single fraction.

Fractions with Variables

Working with fractions that include variables can seem complex at first, but it follows the same basic principles as numerical fractions. When a variable like \( x \) appears in the denominator, it means that the value of \( x \) affects the overall fraction.

• For addition, find a common denominator just as we did without variables, ensuring the operations are performed with respect to \( x \).
• Afterwards, adjust the numerators to reflect their relationship with the new shared denominator, paying attention to keep the expressions involving \( x \) clear and consistent.

In our exercise, despite the presence of \( x \), we found a common denominator \( 15x \) and adjusted numerators, eventually getting to \( \frac{6x + 35}{15x} \). By handling the variable akin to any other number, the problem becomes a straightforward task, providing a complete understanding and solution to adding fractions with variables.