Problem 4 - Elementary Algebra

Question

\(\frac{3 x}{4}+\frac{x}{5}=\frac{3}{10}\)

Step-by-Step Solution

Answer

\(x = \frac{6}{19}\)

Step 1: Identify a Common Denominator

To solve the equation \(\frac{3x}{4} + \frac{x}{5} = \frac{3}{10}\), it is helpful to start by finding a common denominator to eliminate the fractions. The denominators 4, 5, and 10 share a least common multiple (LCM) of 20. We will use 20 as the common denominator.

Step 2: Rewrite Each Fraction with the Common Denominator

Convert each fraction so that they all have the common denominator of 20. This gives us: \(\frac{3x}{4} = \frac{15x}{20}\), \(\frac{x}{5} = \frac{4x}{20}\), and \(\frac{3}{10} = \frac{6}{20}\). The equation now looks like: \(\frac{15x}{20} + \frac{4x}{20} = \frac{6}{20}\).

Step 3: Combine the Terms

Since the fractions now share the same denominator, we can combine the terms on the left side of the equation: \(\frac{15x + 4x}{20} = \frac{6}{20}\) which simplifies to \(\frac{19x}{20} = \frac{6}{20}\).

Step 4: Eliminate the Denominator

To eliminate the denominator, multiply each side of the equation by 20. This gives us: \(19x = 6\).

Step 5: Solve for x

Next, solve for \(x\) by dividing both sides by 19: \(x = \frac{6}{19}\).

Key Concepts

Fractions

Fractions are numbers that represent parts of a whole. They consist of a numerator, the top number, and a denominator, the bottom number. In algebra, fractions often appear in equations and need to be manipulated to simplify the problem.

In the context of solving equations, it is important to understand that each fraction represents a division. For example, \(\frac{3x}{4}\) indicates that the term \(3x\) is being divided into four equal parts.

Fractions can be added, subtracted, or compared when their denominators are the same, allowing us to combine the terms. This is why finding a common denominator is crucial. It simplifies the process of dealing with different fractional parts by unifying the denominators.

Common Denominator

When solving equations with fractions having different denominators, finding a common denominator is essential. The common denominator is simply a number that is a multiple of all the denominators in the equation.

By converting all fractions to have this common denominator, you can easily combine them. In our example exercise, the fractions \(\frac{3x}{4}\), \(\frac{x}{5}\), and \(\frac{3}{10}\) were converted to fractions with a common denominator of 20. Here's how:

• \(\frac{3x}{4}\) becomes \(\frac{15x}{20}\)
• \(\frac{x}{5}\) becomes \(\frac{4x}{20}\)
• \(\frac{3}{10}\) becomes \(\frac{6}{20}\)

This process is known as "clearing the fractions" and it allows us to treat the fractions as regular numbers, combining and simplifying them accordingly.

Solve for x

To solve an equation means to find the value of the variable that makes the equation true. Once the fractions in an equation have a common denominator and have been combined, the equation can be simplified.

For the example equation, \(\frac{15x + 4x}{20} = \frac{6}{20}\) simplifies to \(\frac{19x}{20} = \frac{6}{20}\).
After removing the fractional form by multiplying each side by 20, the equation becomes \(19x = 6\).
The final step is to isolate \(x\) by dividing each side of the equation by 19, resulting in \(x = \frac{6}{19}\).

This process illustrates how converting fractions to have a common denominator is a useful strategy for simplifying and solving fractional equations.