Question 28

Dividing both sides of an equation by the same nonzero number always yields an equivalent equation.

This statement is true as it reflects a fundamental property of equality in algebra. When both sides of an equation are divided by the same nonzero number, the equality is maintained, ensuring that the two sides remain equivalent.

A) Adding a nonzero number or algebraic expression to a given expression always yields an equivalent expression.

This statement is incorrect. Adding a nonzero number or algebraic expression to a single expression does not guarantee an equivalent expression; it alters the original value, thus not maintaining equivalence. An equivalent expression must represent the same value as the original.

B) Dividing both sides of an equation by the same nonzero number always yields an equivalent equation.

This statement is correct. Dividing both sides of an equation by the same nonzero number preserves the equality, meaning that if A = B, then A/c = B/c (where c is a nonzero number) is also true. This property is essential in solving equations without changing their solutions.

C) Multiplying or dividing a given expression by a nonzero number or an algebraic expression always yields an equivalent expression.

This statement is partially incorrect. While multiplying a given expression by a nonzero number maintains equivalence, dividing by a nonzero number does not apply to all expressions uniformly when considering algebraic expressions, as it can lead to undefined scenarios (e.g., division by zero).

D) Adding an algebraic expression to one side of a given equation always yields an equivalent equation.

This statement is incorrect. Adding an algebraic expression to only one side of an equation creates an imbalance unless the same expression is added to both sides. For example, if A = B and you add C to A, you do not have an equivalent equation unless you also add C to B.

Conclusion

Option B is definitively the correct answer because it adheres to the fundamental principles of algebra that govern equality and equivalence in equations. The other options fail to maintain equivalence in their respective operations, either by misapplying algebraic rules or by not considering the necessary conditions for maintaining equality.