Problem 16 - Elementary Algebra

Question

\(\frac{2}{x}+\frac{1}{4}=\frac{13}{20}\)

Step-by-Step Solution

Answer

The solution is \(x = 5\).

Step 1: Eliminate Denominators

To eliminate the fractions, identify the common denominator of the fractions involved: the denominators are \(x\), 4, and 20. The least common multiple for these numbers is 20x. Multiply each term by 20x to eliminate the denominators:\[20x \cdot \frac{2}{x} + 20x \cdot \frac{1}{4} = 20x \cdot \frac{13}{20}\]This simplifies to:\[40 + 5x = 13x\]

Step 2: Solve for x

Rearrange the equation \(40 + 5x = 13x\) to isolate \(x\) on one side. Subtract \(5x\) from both sides:\[40 = 13x - 5x\]This simplifies to:\[40 = 8x\]Divide both sides by 8 to solve for \(x\):\[x = \frac{40}{8}\]Which simplifies to:\[x = 5\]

Key Concepts

Eliminate Denominators in Algebra

Eliminating denominators in algebra is a technique used to simplify equations that contain fractions. The presence of fractions can make solving equations a bit cumbersome. By clearing out the denominators, we transform the equation into a simpler, "denominator-free" form.To eliminate denominators, ask yourself: what is the common denominator for the fractions involved? Identifying this common factor allows you to multiply every term in the equation by it. This way, the fractions transform into whole numbers facilitating easier manipulation.Consider the equation \(\frac{2}{x}+\frac{1}{4}=\frac{13}{20}\). The denominators here are \(x\), 4, and 20. By finding a common factor, which in our case turns out to be \(20x\), you can multiply each component to remove those pesky fractions:

• \(20x \cdot \frac{2}{x} = 40\), and
• \(20x \cdot \frac{1}{4} = 5x\)
• \(20x \cdot \frac{13}{20} = 13x\)

So, it turns our fractional equation into a much simpler \(40 + 5x = 13x\). No more denominators to worry about!

Least Common Multiple

The least common multiple (often abbreviated as LCM) is a crucial concept in mathematics used to find the smallest number that is evenly divisible by all the given numbers. In the context of solving equations with fractions, finding the LCM of the denominators helps us to eliminate them and simplify the equations.For the equation \(\frac{2}{x} + \frac{1}{4} = \frac{13}{20}\), we need to identify the LCM of \(x\), 4, and 20. Since \(x\) is a variable, our LCM will include \(x\), while considering the numeric values of 4 and 20:

• The LCM of 4 and 20: is 20, since 20 is the smallest number that both 4 and 20 divide evenly.
• With \(x\) included, the LCM extends to \(20x\) to ensure every term is covered.

Thus, multiplying the whole equation by \(20x\) allows us to work with a simplified form, removing the need to deal with fractions directly.

Fractional Equations

Fractional equations are equations that feature one or more fractions with variables in the numerator, denominator, or both. Solving them can seem daunting due to the presence of fractions, but using techniques like eliminating denominators and leveraging the least common multiple can make them manageable.The first step in addressing fractional equations like \(\frac{2}{x} + \frac{1}{4} = \frac{13}{20}\) is to recognize the need for a common denominator approach. The goal is to clear the fractions by determining the least common multiple of all denominators and multiplying through.Once denominators are eliminated, the equation becomes more straightforward, allowing for normal algebraic operations to solve for the variable. Here, once the fractions were transformed away, solving \(40 + 5x = 13x\) simply involved rearranging and isolating \(x\) yielding \(x = 5\).By systematically applying these steps, tackling fractional equations becomes a much simpler process.