Florida Teacher Certification Examinations (FTCE) Subject Area Practice Test

In multiplication defined by the equation ab=c, under what condition is b= c-a?

1. c=0

2. b=1

3. c is not equal to zero

4. a=0

The correct answer is: c is not equal to zero

In the equation ab = c, if we rearrange it to express b in terms of a and c, we have b = c/a, provided that a is not equal to zero to avoid division by zero. Now, considering the alternative form b = c - a, we can derive the condition under which this holds true. We need to interpret c - a in relation to ab = c. Rearranging, we assume b equals c - a, leading to a(c - a) = c. This simplifies to ac - a^2 = c. To find when this is true, we can set c(1 - a) = a^2. This introduces the idea that c can vary based on a unless certain conditions apply. The condition for b = c - a to hold can be understood better by analyzing the values of c in relation to a. If c were to equal zero, this would lead to contradictory implications based on values of a and b. Hence, it's necessary for c to maintain a certain value to realize the relationship. Thus, for b = c - a to lead to a consistent scenario within the defined multiplication framework, it should be concluded that c must not equal zero to hold the