Question

complete this assessment to review what youve learned. it will not count toward your grade.

if a system of equations has an infinite number of solutions, what is it called? choose the best answer. enter the number associated with that answer for your response.

1. inconsistent and dependent system of equations
2. consistent system of equations
3. dependent system of equations
4. consistent and dependent system of equations

(1 point)

Explanation:

To determine the answer, we recall the definitions of systems of equations:

• An inconsistent system has no solutions.
• A consistent system has at least one solution.
• A dependent system has infinitely many solutions (since the equations are essentially the same or scalar multiples, leading to coinciding lines/plane etc.).
• A consistent and dependent system is one that is consistent (has solutions) and dependent (infinitely many solutions, as the equations are dependent, meaning one can be derived from the others).

Let's analyze each option:

1. "inconsistent and dependent" is incorrect because inconsistent systems have no solutions, so they can't be dependent (which requires solutions).
2. "consistent system" is too broad; a consistent system could have one solution (independent) or infinitely many (dependent).
3. "dependent system" is part of the correct description but doesn't specify it's consistent (though by definition, a dependent system is consistent as it has solutions). However, the most accurate term is "consistent and dependent" because it specifies both that there are solutions (consistent) and infinitely many (dependent).
4. "consistent and dependent system of equations" is correct because a system with infinite solutions is consistent (has solutions) and dependent (the equations are dependent, leading to infinite solutions).

Answer:

1. consistent and dependent system of equations