Problem 80 - College Algebra

Question

Suppose that you solve \(\frac{x}{5}-\frac{x}{2}=1\) by multiplying both sides by \(20,\) rather than the least common denominator of 5 and 2 (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

Multiplying the equation by 20 gives the same solution for x, which is \(-\frac{10}{3}\). Using the least common denominator is a practical choice for clearing the equation of fractions, but not the only choice.

Step 1: Multiply each side of the equation by 20

The equation is given as \(\frac{x}{5}-\frac{x}{2}=1\). Start by multiplying both sides of the equation by 20. This gives \(20\cdot\frac{x}{5} - 20\cdot\frac{x}{2} = 20\cdot1\). This simplifies to \(4x - 10x = 20\).

Step 2: Solve the equation

Combine like terms to simplify the left side of the equation: \(4x - 10x = -6x\). Therefore, the equation is \(-6x = 20\). Finally, solve for \(x\) by dividing both sides of the equation by \(-6\). This tells us that \(x = -\frac{20}{6} = -\frac{10}{3}\).

Step 3: Discussion on the use of the lowest common denominator

The same value for \(x\) should be obtained when multiplying the equation by the lowest common denominator, 10. When you multiply by the least common denominator, you ensure that you not only eliminate the fractions but also do not change the problem's value. Using a factor of the least common denominator, like 20, also clears the fractions and still delivers the correct answer, but the equation becomes weighted with larger numbers, which might make it more difficult to solve in some cases. Thus, using the least common denominator is a practical choice rather than a necessary one.

Key Concepts

Least Common Denominator

When solving equations that involve fractions, finding the least common denominator (LCD) is a helpful technique.
The least common denominator is the smallest number that each of the denominators can evenly divide into. In the equation \(\frac{x}{5} - \frac{x}{2} = 1\), the denominators are 5 and 2. The least common multiple of these numbers is 10.
Using the LCD allows you to clear the fractions gracefully without making the numbers larger and more complex than necessary.
Even though multiplying by any common multiple (like the example's 20) can clear the fractions, it may make the calculation more cumbersome.
Choosing the LCD simplifies the task, making the arithmetic less prone to errors and calculations easier to manage.

Clearing Fractions

Clearing fractions is an important step when working with equations involving fractional terms.
It involves multiplying both sides of the equation by a number that will eliminate the denominators, turning the fractions into whole numbers.
In the original problem, multiplying by 20 cleared the fractions by removing the denominators 5 and 2.
This effectively transformed the equation from \(\frac{x}{5} - \frac{x}{2} = 1\) to \(4x - 10x = 20\).
By clearing fractions, you reduce complexity and make the remaining equation much simpler to handle.
Once the fractions are cleared, you can focus on solving the equation using basic algebraic techniques without being distracted by fractional arithmetic.
This technique not only simplifies solving the equation but also helps in better understanding and visualizing the problem.

Rational Equations

Rational equations are equations that involve one or more fractions with variables in the numerator or the denominator.
Solving these types of equations requires careful handling of both fractions and the algebraic manipulations that follow.
The provided exercise \(\frac{x}{5} - \frac{x}{2} = 1\) is an example of a rational equation.
To solve it, you must first clear the fractions, often by using the least common denominator.
Rational equations can be tricky because they sometimes present extraneous solutions, which are incorrect answers that appear valid in the transformed equation but do not satisfy the original equation.
Therefore, it's crucial to check your solution by plugging it back into the original equation to ensure it holds true.
This double-checking step confirms the accuracy of your solution and ensures that the restructuring of the equation didn't alter the original problem's validity.