Problem 108 - College Algebra Essentials

Question

Suppose that you solve \(\frac{x}{5}-\frac{x}{2}=1\) by multiplying both sides by 20 rather than the least common denominator (namely, 10 ). Describe what happens. If you get the correct solution, why do you think we clear the equation of fractions by multiplying by the least common denominator?

Step-by-Step Solution

Answer

The solution for x is -10/3. Using the LCD, in general, simplifies the equation, allowing for a solution to be found more easily and with fewer potential errors.

Step 1: Multiply the equation by 20

The original equation is \(\frac{x}{5} - \frac{x}{2} = 1\). Multiplying every fraction by 20 to remove the denominator, gives 4x - 10x = 20.

Step 2: Simplify the equation

Simplify the left-hand side by subtracting 10x from 4x, which gives you -6x = 20.

Step 3: Solve for x

After, divide both sides of the equation by -6 to solve for x: so \(x = \frac{20}{-6} = -\frac{10}{3}\).

Step 4: Explanation of using the least common denominator

The solution obtained is correct, even when not using the LCD. However, applying the LCD is a simpler and more common method because it reduces the complexity of the equation, thus calculating the answer with fewer steps and less chance of making calculation errors. This is more crucial for equations with more fractions and larger denominators.

Key Concepts

Multiplying by Least Common Denominator

When solving equations with fractions, one effective strategy to simplify the process is multiplying by the least common denominator (LCD). The LCD is the smallest number that each denominator in the equation can divide into evenly. Using this method can make the equation much easier to handle. For example, in an equation like \(\frac{x}{5} - \frac{x}{2} = 1\), the denominators are 5 and 2.
The least common multiple of these numbers is 10, which is the LCD.

Here's why multiplying by the LCD is advantageous:

• It removes the fractions entirely, making the equation more straightforward.
• After eliminating the fractions, you're left with simpler whole number terms.
• Working with whole numbers decreases the chance of arithmetic errors.

This method is particularly beneficial when dealing with multiple fractions or when the denominators are larger integers.
By simplifying early, you'll save time and reduce possible mistakes in your computations.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, or the denominator, or both contain algebraic expressions. Solving linear equations that involve algebraic fractions can be seen as tricky at first, but understanding the underlying concepts can make the process more intuitive.
An equation like \(\frac{x}{5} - \frac{x}{2} = 1\) contains algebraic fractions because of the variable \(x\) in the numerators. The primary goal when dealing with algebraic fractions is to remove those fractions to simplify your solving process.

Consider these steps when working with algebraic fractions:

• Identify all fractions in the equation.
• Determine the least common denominator (LCD) of these fractions.
• Multiply every term in the equation by the LCD to clear the fractions.

Once fractions are eliminated, proceed to solve the equation as you would with any other linear equation. This results in a clearer path to finding the variable's value.

Simplifying Equations

Simplifying equations is a crucial part of solving math problems efficiently. Once you multiply the entire equation by the least common denominator (LCD), your next goal is to simplify the resultant equation.
Take the equation \(\frac{x}{5} - \frac{x}{2} = 1\). After multiplying by the LCD of 10, you get \(2x - 5x = 10\).

Simplification involves these key steps:

• Combine like terms on either side of the equation.
• Adjust the equation so that one side has the variable terms and the other side has the constants.
• Use simple arithmetic to solve for the variable.

In our example, simplifying by combining \(2x - 5x\) results in \(-3x\), and then solving \(-3x = 10\) gives us \(x = -\frac{10}{3}\).
Simplifying equations helps turn complex problems into more manageable ones, allowing you to focus on the logic rather than lessing details. It's a valuable skill that encourages accuracy and confidence in solving even challenging problems.