Question

Let a,b,c, and d be real numbers. Given that ac=1, db+c is undefined, and abc=d, which of the following must be true? A. a=0 or c=0 B. a=1 and c=1 C. a=−c D. b=0 E. b+c=0

Answer 1

The correct statement is that b+c=0.

Step-by-step explanation:

To determine which of the given options must be true, we will analyze each statement using the given information:

A. a=0 or c=0: This statement is not necessarily true. We know that ac=1, but this does not imply that either a or c must be 0.

B. a=1 and c=1: This statement is not correct. From the given information, we know that abc=d, but we cannot conclude that a=b=c=1.

C. a=-c: This statement is not necessarily true. From the given information, we know that abc=d, but we cannot conclude that a=-c.

D. b=0: This statement is not necessarily true. From the given information, we know that db+c is undefined, but this does not imply that b=0.

E. b+c=0: This statement must be true. From the given information, we know that db+c is undefined, which means that b+c cannot have a real value. Therefore, b+c=0.

So, the correct answer is option E.

Answer 2

After evaluating the conditions ac=1, db+c is undefined, and abc=d, the only statement that must be true is B. a=1 and c=1, as this satisfies all given conditions without any contradiction.

Step-by-step explanation:

The question provided is asking to determine a mathematical relationship given specific conditions for the real numbers a, b, c, and d. We've been given three pieces of information: ac = 1, db + c is undefined, and abc = d. Let's analyze the given conditions step by step.

• Since ac = 1, and we are working with real numbers, neither a nor c can be zero because the product of two real numbers is 1 only if both numbers are either positive or negative reciprocals of each other. Therefore, A. a=0 or c=0 is not true.
• From ac = 1, if we assume either a or c is 1, the other must also be 1 to satisfy the equation. Thus, B. a=1 and c=1 is a possibility and is true.
• Option C. a=−c cannot be true, as this would result in ac being -1, which contradicts ac = 1.
• Since db + c is undefined, and we know c is defined because it has a value that makes ac = 1, b must be the undefined term, which means that D. b=0 is not necessarily true as zero is defined.
• It's the term db that's undefined, not b, which implies that d must be undefined. We were also given abc = d; since we know a and c are real numbers, this implies b must be the undefined term. Therefore, E. b+c=0 is also not necessarily true.

Therefore, the only statement that must be true given the conditions is B. a=1 and c=1, which satisfies ac=1 and abc=d, with d also being equal to 1 based on the given conditions.