Problem 25 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ \frac{x}{2}-\frac{x}{3}=4 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 24 \).

Step 1: Find the Least Common Denominator

Identify the denominators in the equation, which are 2 and 3. The least common denominator (LCD) for 2 and 3 is 6.

Step 2: Eliminate the Fractions

Multiply every term in the equation by the LCD to eliminate fractions: \( 6 \left( \frac{x}{2} \right) - 6 \left( \frac{x}{3} \right) = 6 \times 4 \). This simplifies to \( 3x - 2x = 24 \).

Step 3: Simplify the Equation

Combine like terms: \( 3x - 2x = x \). This results in the simplified equation \( x = 24 \).

Key Concepts

Least Common Denominator

When dealing with fractions in linear equations, finding the least common denominator (LCD) is crucial in simplifying the problem. The LCD of two or more fractions is the smallest number that all the denominators can divide into without leaving a remainder.
For example, in the equation \( \frac{x}{2} - \frac{x}{3} = 4 \), the denominators are 2 and 3. To solve this equation effectively, we need to determine the LCD of these fractions.
The multiples of 2 are 2, 4, 6, 8, and so on, while the multiples of 3 are 3, 6, 9, and 12. The LCD is the smallest number that appears in both lists, which is 6 in this case. Identifying the LCD allows you to harmonize the equation, making the next steps significantly more straightforward.

Eliminate Fractions

Eliminating fractions in an equation makes it easier to solve. Once you have identified the least common denominator, you can multiply every term in the equation by this value.
In our example, the LCD is 6. By multiplying each term by 6, we can effectively remove the fractions. Let's see how it works:

• \(6 \times \frac{x}{2} = 3x\)
• \(6 \times \frac{x}{3} = 2x\)
• \(6 \times 4 = 24\)

Rewriting the equation, we now have a fraction-free equation: \(3x - 2x = 24\).
By eliminating fractions, we simplify the equation into a more manageable linear form.

Combine Like Terms

Once fractions are eliminated, you often need to simplify the equation by combining like terms. Like terms are terms that have the same variable raised to the same power.
Let's look at the equation we simplified earlier: \(3x - 2x = 24\). Notice that \(3x\) and \(2x\) are like terms because they both have the variable \(x\).
Combining them is straightforward:

• Subtract \(2x\) from \(3x\), which gives us \(x\).

Now, the equation is simple: \(x = 24\).
By combining like terms, you reduce complexity and isolate the variable, making it easy to pinpoint the solution.