Assessment and Learning in Knowledge Spaces (ALEKS) Practice Exam

In the equation Ax + By = C, what do A and B need to fulfill?

1. Must be equal to zero

2. Both must be negative

3. Cannot both be zero

4. Must be whole numbers

The correct answer is: Cannot both be zero

In the equation Ax + By = C, A and B serve as the coefficients that determine the slope and position of the line represented by the equation in a two-dimensional Cartesian coordinate system. For the equation to represent a valid line, the conditions related to A and B must ensure that the line has a defined slope. The requirement that A and B cannot both be zero is crucial because if both coefficients were zero, the equation would simplify to 0 = C. If C is not zero, this would represent an inconsistency (no solutions), and if C equals zero, it would imply that any x and y values would have to satisfy 0 = 0, leading to an indeterminate situation rather than a specific line. Thus, at least one of the coefficients must be non-zero to maintain the integrity of the linear equation, allowing it to signify a unique line in the coordinate plane. The other options suggest conditions about the values of A and B that are not necessary for the equation to hold. For example, A and B do not need to be equal to zero or both negative, nor do they need to be whole numbers, as the coefficients can be any real numbers, providing a wider range of possibilities for the equation.