Difficulty: Hard

Average Score: 33%

Which of the following statements is true?

Rationale

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

When both sides of an equation are divided by the same nonzero number, the equality of the two sides is maintained, ensuring that the resulting equation remains valid and equivalent to the original.

A (Adding a nonzero number or algebraic expression to a given expression always yields an equivalent expression) YOUR ANSWER

This statement is incorrect because adding a nonzero number or algebraic expression to one side of an equation without making the same addition to the other side does not preserve the equality. Therefore, the expressions would not be equivalent.

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

This choice is misleading. While multiplying an expression by a nonzero number does yield an equivalent expression, dividing an expression by a number does not apply to the entire expression unless it is done uniformly to both sides of an equation. Thus, this statement does not hold true universally.

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

This statement is false as well. Similar to option A, if an algebraic expression is added only to one side of an equation without making the same addition to the opposite side, the equation will no longer be equivalent, violating the principle of equality.

Conclusion

In algebra, maintaining the integrity of equations is crucial for accurate problem-solving. Dividing both sides of an equation by the same nonzero number is a valid operation that upholds the equation's equivalence. In contrast, adding or multiplying without consistent application to both sides can alter the equality, highlighting the importance of careful manipulation of equations.

Related Questions

View all