Problem 6 - Elementary and Intermediate Algebra

Question

A student was asked to solve a rational equation. The first step of his solution is as follows: $$ 12 x\left(\frac{5}{x}+\frac{2}{3}\right)=12 x\left(\frac{7}{4 x}\right) $$ a. What equation was he asked to solve? b. What LCD is used to clear the equation of fractions?

Step-by-Step Solution

Answer

a. \( \frac{5}{x} + \frac{2}{3} = \frac{7}{4x} \) b. LCD is \(12x\).

Step 1: Identify the Original Equation

Given the equation \(12x\left(\frac{5}{x}+\frac{2}{3}\right)=12x\left(\frac{7}{4x}\right)\), we see that the student is likely trying to clear fractions from a rational equation with an original form similar to \( \frac{5}{x} + \frac{2}{3} = \frac{7}{4x} \).

Step 2: Determine the LCD

To clear the equation of fractions, we need to find the Least Common Denominator (LCD) of the denominators \(x\), \(3\), and \(4x\). The LCD must be a multiple of all these denominators. Since \(x\) and \(4x\) share \(x\), the LCD is \(12x\).

Step 3: Formulate the Cleared Equation

By multiplying all terms of \(\frac{5}{x} + \frac{2}{3} = \frac{7}{4x}\) by \(12x\), we can see the transformation leading to \(12x\left(\frac{5}{x}\right)+12x\left(\frac{2}{3}\right)=12x\left(\frac{7}{4x}\right)\).

Key Concepts

Least Common Denominator

Dealing with fractions often requires them to have a common baseline or bottom value, known as the denominator. The Least Common Denominator (LCD) is the smallest multiple that all denominators in a set of fractions can divide into without leaving a remainder. This is crucial in rational equations, such as the one given by the student: \(\frac{5}{x} + \frac{2}{3} = \frac{7}{4x}\).To find the LCD, identify all unique denominators present in the equation. In this example, the denominators are \(x\), \(3\), and \(4x\). Since \(4x\) already includes \(x\), we consider the least common multiple of the numbers \(3\) and \(4\), which is \(12\). Multiply this by \(x\) because it must also account for \(x\) shared by \(x\) and \(4x\). Therefore, the LCD is \(12x\). Having an LCD allows us to combine terms more easily.

Fractions

A fraction represents a part of a whole and consists of two parts: a numerator (top number) and a denominator (bottom number). The fraction \(\frac{a}{b}\) signifies "\(a\) parts out of \(b\)." In rational equations, different fractions are often involved. For example, in the rational equation \(\frac{5}{x} + \frac{2}{3} = \frac{7}{4x}\), different unknowns and constants signify different portions of an overall equation. When solving rational equations, it is vital to handle each fraction carefully. Often, with different denominators, combining them directly is challenging. Instead, finding a common ground or a Least Common Denominator helps simplify the process. Once the fractions are over a common denominator, they can be easily added, subtracted, or transformed as needed. This step is part of making solving these equations manageable, and it's an essential competence when handling more complex algebraic expressions.

Clearing Fractions

Clearing fractions from an equation makes it easier to work with because it leaves a polynomial equation, which is simpler to manage. When dealing with the equation \(\frac{5}{x} + \frac{2}{3} = \frac{7}{4x}\), clearing fractions is accomplished by multiplying every term by the least common denominator, which was found to be \(12x\) in this case.By multiplying each term by the LCD, the denominators are effectively eliminated. Here’s how it works:

• \(12x \times \frac{5}{x} = 60\)
• \(12x \times \frac{2}{3} = 8x\)
• \(12x \times \frac{7}{4x} = 21\)

Once all fractions are cleared, the equation simplifies to a form like \(60 + 8x = 21\), which can be solved for the variable without dealing with fractions. This simplification is crucial, as it reduces potential errors and eases the solution process.