Problem 95 - Introductory Algebra for College Students

Question

Suppose that you solve \(\frac{x}{5}-\frac{x}{2}=1\) by multiplying both sides by \(20,\) rather than the least common denominator, \(10 .\) 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 to the equation \(\frac{x}{5}-\frac{x}{2}=1\) when multiplied by \(20\) (instead of the least common denominator) is \(x = -\frac{10}{3}\). Even though a non-standard common denominator was used, the correct solution was still obtained. We usually multiply by the least common denominator to simplify the equation however, using a larger common denominator does not affect the final solution.

Step 1: Find the common factor

First, find the common factor between 5 and 2, which is 20. Even though the least common factor between 2 and 5 is 10, the instruction has asked to use 20. So we would be multiplying every term of the equation by 20.

Step 2: Multiply each term

Multiply each term in the equation by 20. \((20*\frac{x}{5})-(20*\frac{x}{2}) = 20*1\), this simplifies to \(4x-10x = 20\).

Step 3: Solve the simplified equation

Now, solve the simplified equation. \(4x-10x = 20\) simplifies further to \(-6x = 20\).

Step 4: Find the final answer

Isolate x in the equation \(-6x = 20\) by moving \(6x\) to the other side of the equation. \(x = \frac{-20}{6} = -\frac{10}{3}\)

Step 5: Reflect on the process

The least common denominator was not used in this instance, instead a number larger than the least common denominator (20 instead of 10) was used. The resulting solution is identical to the solution obtained when the least common denominator was used.