What does multiplying both sides of an equation by a variable imply?
A) The operation is always reversible
C) The result is always a valid solution
D) None of the above
B) It can lead to extraneous solutions
The correct answer and explanation is:
The correct answer is B) It can lead to extraneous solutions.
When you multiply both sides of an equation by a variable, you introduce the potential for extraneous solutions, particularly when the variable could be zero. If the variable is zero, multiplying both sides by zero can create a false solution, even if it satisfies the modified equation. This is a critical issue when solving equations, as you may inadvertently generate solutions that don’t work in the original equation. This is especially true in rational expressions or equations involving square roots.
For example, consider the equation 1x=2\frac{1}{x} = 2. If you multiply both sides by xx, you get 1=2×1 = 2x, which simplifies to x=12x = \frac{1}{2}. However, you must check if this solution satisfies the original equation, as the operation of multiplying by xx implicitly assumes x≠0x \neq 0. If x=0x = 0, the original equation would be undefined, and multiplying both sides by zero would lead to an invalid solution.
This issue arises because multiplying both sides by a variable can change the domain of the equation. When you perform such an operation, it is essential to verify that the solutions satisfy the original equation, especially in cases where division by the variable or roots of the variable are involved.
In conclusion, multiplying both sides of an equation by a variable can indeed lead to extraneous solutions, and such solutions need to be checked by substituting back into the original equation to verify their validity.