Problem 98 - Introductory Algebra for College Students

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Although I can solve \(3 x+\frac{1}{5}=\frac{1}{4}\) by first subtracting \(\frac{1}{5}\) from both sides, I find it easier to begin by multiplying both sides by \(20,\) the least common denominator.

Step-by-Step Solution

Answer

Both the statement makes sense and doesn't make sense. It makes sense as both methods result in the correct answer and beginning by multiplying by the LCD is a valid approach. It doesn't make sense claiming that it is 'easier' as it involves more operations compared to the first method of subtracting from both sides.

Step 1: Method 1: Subtracting from Both Sides

In this approach, subtract \(\frac{1}{5}\) from both sides of the equation \(3x + \frac{1}{5} = \frac{1}{4}\). This simplifies the equation to \(3x = \frac{1}{4} - \frac{1}{5}\). Which when solved, yields \(x \approx 0.033\).

Step 2: Method 2: Multiplying Both Sides by LCD

In this approach, multiply both sides of the equation by 20, the LCD of \(5\) and \(4\). Therefore, \(20 \cdot (3x + \frac{1}{5}) = 20 \cdot \frac{1}{4}\) simplifies to \(60x + 4 = 5\). When this equation is solved, \(x \approx 0.033\).

Step 3: Comparison and Analysis

Both methods are valid approaches to solving the equation and they yield the same result. However, the second method involves more operations (multiplication before subtraction) than the first method (just subtraction). Therefore, it can be reasoned that the first method might be easier or simpler to use in this particular case.

Key Concepts

Algebraic Fractions

Understanding algebraic fractions is crucial when working with equations that involve fractions. An algebraic fraction is simply a fraction where the numerator, the denominator, or both contain algebraic expressions. For instance, in the equation \(3x + \frac{1}{5} = \frac{1}{4}\), there are fractions with numerators and denominators that have constants and variables.

When dealing with such fractions, operations like addition, subtraction, multiplication, and division can be applied, just as with numerical fractions. However, the presence of variables means that finding a common denominator can assist in simplifying the equation. This simplification process often makes it easier to solve for the variable in question.

Least Common Denominator

The least common denominator (LCD) is the smallest number that can be divided evenly by all denominators in the equation. In our example, where \(3x + \frac{1}{5} = \frac{1}{4}\), the denominators are 5 and 4. The LCD for 5 and 4 is 20 because it is the smallest number that both 5 and 4 can divide into without leaving a remainder.

Using the LCD to clear fractions from an equation is a powerful technique that can simplify problem solving. By multiplying each term of the equation by the LCD, you convert the equation into one without fractions, which is often easier to manage. This is particularly helpful when dealing with equations involving algebraic fractions, because it transforms the equation into a simpler form in which standard linear equation solving techniques can be readily applied.

Equation Solving Techniques

There are multiple techniques for solving linear equations, and choosing the best method can make the process more efficient. Common strategies include:

• Isolation of the variable: Rearrange the equation to get the unknown variable by itself on one side of the equation.
• Combining like terms: Simplify the equation by adding or subtracting terms with the same variable.
• Using the distributive property: Apply this property to remove parentheses in an expression.
• Clearing fractions: Multiply every term by a common denominator to get rid of fractions, as seen with the LCD technique.

Each of these techniques may be used independently or in combination to solve a linear equation. The choice of method can be influenced by the particular structure of the equation and the preference of the individual solving it.

Comparing Solution Methods

When solving an equation, such as \(3x + \frac{1}{5} = \frac{1}{4}\), it's beneficial to compare different solution methods to determine which is more efficient or easier to perform. In the given exercise, we have two methods:

• Subtracting from both sides: This technique simplifies the equation directly by combining like terms.
• Multiplying both sides by the LCD: This method clears the fractions first, which results in a different but equivalent equation to solve.

Each method has advantages and potential drawbacks depending on the complexity of the equation and the skill level of the student. While some may prefer the directness of subtraction, others might find that dealing with whole numbers is easier after using the LCD. Analyzing the steps required and the resulting equations can help students decide which method aligns with their strengths and understanding.