Problem 98 - Introductory and Intermediate Algebra for College Students 4th

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

The statement 'makes sense'. Multiplying each side of the equation by the least common denominator to begin with can simplify the process, especially for those who prefer not to work with fractions. It is alternative and acceptable way to solve the equation, however subtracting \(\frac{1}{5}\) from both sides first is equally valid and can be easier for some.

Step 1: Evaluate the original equation

Consider the original equation \(3x + \frac{1}{5} = \frac{1}{4}\). The aim is to isolate x to solve the equation, which often involves simplifying the equation.

Step 2: Assess the proposed operation

The proposed operation is multiplying both sides by 20, the least common denominator of 5 and 4 (the denominators in the equation). If we perform this operation, we get \(20*(3x + \frac{1}{5}) = 20*\frac{1}{4}\), which simplifies to \(60x + 4 = 5\). Note that this operation eliminates the fractions, which can make the equation easier to solve.

Step 3: Compare the proposed method with the alternative

An alternative operation is subtracting \(\frac{1}{5}\) from both sides first as hinted at in the statement. This gives us \(3x = \frac{1}{4} - \frac{1}{5}\), which simplifies to \(3x = \frac{1}{20}\). While it's not incorrect, this method requires dealing with fractions, which some people might find more challenging.

Key Concepts

Equation Solving

When it comes to solving equations, the main goal is to find the value of the unknown variable, often denoted as \(x\) or \(y\). This involves performing a series of operations that maintain the equality of both sides of the equation.

Start by simplifying both sides of the equation if necessary. This might involve distributing or combining like terms.

Next, use operations such as adding, subtracting, multiplying, or dividing to isolate the variable on one side of the equation.

In our case, we started with the equation \(3x + \frac{1}{5} = \frac{1}{4}\). The ultimate goal was to isolate \(x\), leading us to use operations like multiplication to simplify the process.
Choosing the right operation depends on the equation's form and the involved numbers. As you'll see, understanding fractions and how to manipulate them can be crucial in equation solving.

Fractions

Fractions represent a part of a whole. In algebra, they often appear in equations and can pose a challenge if one is not familiar with handling them. Let's break it down simply.

A fraction consists of a numerator (top number) and a denominator (bottom number).

The key operations with fractions include addition, subtraction, multiplication, and division.

In our exercise, we're dealing with fractions: \(\frac{1}{5}\) and \(\frac{1}{4}\).
To solve the equation efficiently, it helps to be comfortable with fractions so that operations like finding a common denominator or performing arithmetic with fractions become manageable.

Least Common Denominator

Finding the least common denominator (LCD) is a critical skill when dealing with fractions in equations. The LCD is the smallest multiple that two or more denominators share.

It simplifies the task of adding, subtracting, or comparing fractions.

In our equation, the denominators were 5 and 4. The LCD of these numbers is 20 because it's the smallest number that both 5 and 4 can divide without leaving a remainder.

Multiplying each side of the equation by the LCD helps to eliminate fractions altogether, which simplifies the solving process.

Applying this concept to our equation made it straightforward by transforming the fraction-laden equation to \(60x + 4 = 5\), a much easier form to handle.

Isolating Variable

The concept of isolating the variable is a central aspect of solving equations. Once the variable is isolated, the equation effectively tells us what number it equals.

The process involves rearranging the equation so that the variable stands alone on one side of the equality.

In the exercise, post multiplication by 20, our equation was \(60x + 4 = 5\). To isolate \(x\), we followed these steps:
1. Subtract 4 from both sides to get \(60x = 1\). 2. Divide by 60 to solve for \(x\), yielding \(x = \frac{1}{60}\).
This step-by-step simplification is typical in algebraic problems and helps to ultimately uncover the value of the variable sought.