When a system of two linear equations is consistent, which of the following must be true?
Its solution is the empty set.
Its solution set consists of at least one ordered pair.
Its solution set consists of exactly one ordered pair.
Its solution set consists of infinitely many ordered pairs.
To determine which statement must be true when a system of two linear equations is consistent, we need to understand what it means for a system of equations to be consistent.
A system of two linear equations is considered consistent if there is at least one solution that satisfies both equations simultaneously. In other words, the equations have common solutions and intersect at one or more points.
Now let's consider the options:
1. "Its solution is the empty set": This statement implies that there are no solutions to the system of equations. This would mean that the equations do not intersect and are parallel or coincident lines. However, this contradicts the assumption that the system is consistent, so this statement is not true.
2. "Its solution set consists of at least one ordered pair": This statement correctly reflects the definition of a consistent system. If there is at least one ordered pair that satisfies both equations, then the system is indeed consistent.
3. "Its solution set consists of exactly one ordered pair": This statement indicates that there is only one solution to the system of equations. Consistent systems can indeed have exactly one solution, so this statement is also true.
4. "Its solution set consists of infinitely many ordered pairs": This statement describes an inconsistent system rather than a consistent one. An inconsistent system means that the equations do not have any common solutions and do not intersect. Therefore, this statement is not true for a consistent system.
In summary, the statements that must be true for a system of two linear equations to be consistent are:
- "Its solution set consists of at least one ordered pair."
- "Its solution set consists of exactly one ordered pair."