Problem 4 - Introductory Algebra for College Students

Question

Solve each rational equation. $$\frac{5 x}{4}=\frac{x}{12}-\frac{x}{2}$$

Step-by-Step Solution

Answer

Based on the solution steps, the solution to the equation \(\frac{5x}{4}=\frac{x}{12}-\frac{x}{2}\) is \(x = 0\).

Step 1: Eliminate the Fractions

To solve the equation and remove fractions, we multiply each term by the least common multiple (LCM) of the denominators (4, 12, and 2). The LCM is 12. Multiplying each term by 12 gives: \(15x = x - 6x\)

Step 2: Simplify the Equation

Combine like terms on the right side of the equation to simplify: \(15x = -5x\).

Step 3: Solve for x

In order to get \(x\) alone on one side, add \(5x\) to both sides: \(20x = 0.\) Divide both sides by 20 to solve for \(x\): \(x = 0.\)

Key Concepts

Understanding the Least Common Multiple

When solving rational equations, involving fractions, the least common multiple (LCM) becomes essential. The LCM of a set of numbers is the smallest number that all the numbers can divide without leaving a remainder. By finding the LCM, you can eliminate the fractions by multiplying each term by this number.

In the given problem, the denominators are 4, 12, and 2. To find the LCM:

• List the multiples of each number until you find a common one.
• The multiples of 4: 4, 8, 12, 16, ...
• The multiples of 12: 12, 24, 36, ...
• The multiples of 2: 2, 4, 6, 8, 10, 12, ...

The smallest common multiple is 12. This LCM is used to eliminate fractions by multiplying each term in the equation, simplifying the arithmetic and making the problem more manageable.

Multiplying by the LCM transitions fractions into whole numbers, paving the way for easier manipulation of the equation.

The Art of Combining Like Terms

After removing fractions, the next step in solving equations is typically to simplify by combining like terms. Like terms are terms in an equation that have identical variable parts. For example, in the equation from the solution, we started with:

• \(15x = x - 6x\)

Here, \(x\) and \(-6x\) are like terms because they both contain the variable \(x\). Combining them makes the equation simpler by reducing the number of terms you need to consider.

To simplify:

• Add or subtract the coefficients (numbers in front of the variables) of like terms together.
• This yields \(15x = -5x\) which consolidates the equation.

Combining like terms helps reduce complexity and focus on one aspect of the variable, making it easier to solve the equation.

Solving for the Variable

Once the equation is simplified, the last step is to solve for the variable. Solving for a variable means isolating that variable on one side of the equation to determine its value. In this exercise, we needed to solve for \(x\).

From:

• \(15x = -5x\)

We want \(x\) alone on one side. By adding \(5x\) to both sides:

• \(15x + 5x = -5x + 5x\)
• This simplifies to \(20x = 0\)

After isolating \(x\), divide both sides by 20 to find the value of \(x\):

• \(x = 0\)

Solving for the variable is a crucial step in algebra as it allows us to find specific values that satisfy the given equation.