Problem 20 - Elementary and Intermediate Algebra

Question

Solve each equation and check the result. If an equation has no solution, so indicate. $$ \frac{x}{4}=\frac{1}{2}-\frac{3 x}{20} $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{5}{4} \).

Step 1: Eliminate Fractions by Finding a Common Denominator

The equation given is \( \frac{x}{4} = \frac{1}{2} - \frac{3x}{20} \). To eliminate the fractions, we find the least common denominator (LCD) of 4, 2, and 20, which is 20. Multiply every term by 20 to clear the fractions: \[ 20 \times \frac{x}{4} = 20 \times \frac{1}{2} - 20 \times \frac{3x}{20} \] This simplifies to: \[ 5x = 10 - 3x \]

Step 2: Move All Terms Involving x to One Side

Now that we have \( 5x = 10 - 3x \), we need to get all terms with \( x \) on one side of the equation. Add \( 3x \) to both sides: \[ 5x + 3x = 10 - 3x + 3x \] which simplifies to: \[ 8x = 10 \]

Step 3: Solve for x

With \( 8x = 10 \), divide both sides by 8 to isolate \( x \): \[ x = \frac{10}{8} \] Simplify the fraction by dividing the numerator and the denominator by 2: \[ x = \frac{5}{4} \]

Step 4: Check the Solution

Substitute \( x = \frac{5}{4} \) back into the original equation to check the solution: \[ \frac{x}{4} = \frac{1}{2} - \frac{3x}{20} \] Substitute \( x \): \[ \frac{5/4}{4} = \frac{1}{2} - \frac{3(5/4)}{20} \] Simplify each term: \[ \frac{5}{16} = \frac{1}{2} - \frac{15}{80} \] Convert \( \frac{1}{2} \) to have the same denominator as \( \frac{15}{80} \): \[ \frac{40}{80} - \frac{15}{80} = \frac{25}{80} \] and this simplifies to \( \frac{5}{16} \), confirming that \( x = \frac{5}{4} \) is correct.

Key Concepts

Fraction Elimination

Understanding how to eliminate fractions is a fundamental skill in solving equations. Often, fractions make equations seem complex, but we can simplify them using a straightforward technique: clearing fractions through multiplication.

Here's how it works:

• Identify the denominators in the equation.
• Find the least common denominator (LCD) of these fractions. It’s the smallest multiple that can be divided evenly by each of the denominators.
• Multiply every term in the equation by the LCD. This will eliminate the fractions, transforming the equation into a simpler form with whole numbers.

In our initial problem, the LCD of 4, 2, and 20 is 20. By multiplying every term by 20, all fractions are cleared, leading to a simpler equation without fractions. This step is crucial as it makes the subsequent algebraic manipulation much more straightforward.

Common Denominator

Finding a common denominator is essential when dealing with multiple fractions. This process involves aligning all the fractions so they share the same denominator, which allows for easy manipulation and simplification.

Steps to find a common denominator include:

• Identify all denominators in the fractions you are working with.
• Calculate the least common multiple (LCM) of these denominators, which will be your common denominator.
• Convert each fraction to an equivalent fraction with the common denominator. This often requires multiplying both the numerator and the denominator of each fraction by a necessary factor.

In the original equation, we needed to bring all terms to have a denominator of 20. This ensured each term could be easily evaluated, reduced, or cleared without altering the equation's balance. Aligning denominators not only maintains equation integrity but also facilitates the next steps in solving the equation.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form, where the numerator and the denominator have no common factors other than 1.

Steps to simplify fractions:

• Determine the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCF.

This results in a reduced or simplified fraction, where it's easier to see the relationship between the numerator and the denominator.

In our problem, after solving for \( x \), we obtained \( \frac{10}{8} \). We noticed that the numerator and the denominator can both be divided by their GCF, which is 2. After dividing, we get \( \frac{5}{4} \), which is the simplified form. Simplifying fractions is a crucial step in presenting your final answer clearly and correctly, ensuring it's in the best form for interpretation and further use.