Questions: A system of linear equations with more equations than unknowns is sometimes called an overdetermined system. Can such a system be consistent? Illustrate your answer with a specific system of three equations in two unknowns. Choose the correct answer below. A. No, overdetermined systems cannot be consistent because there are no free variables. For example, the system of equations below has no solution. x1=2, x2=4, x1+x2=24 C. Yes, overdetermined systems can be consistent. For example, the system of equations below is consistent because it has the solution (Type an ordered pair.) x1=2, x2=4, x1+x2=6 B. Yes, overdetermined systems can be consistent. For example, the system of equations below is consistent because it has the solution (Type an ordered pair.) x1=2, x2=4, x1+x2=8 D. No, overdetermined systems cannot be consistent because there are fewer free variables than equations. For example, the system of equations below has no solution. x1=2, x2=4, x1+x2=12

Solution Steps

An overdetermined system is a system of linear equations where there are more equations than unknowns. Such systems can sometimes be consistent, meaning they have at least one solution, but they are often inconsistent, meaning they have no solution.

We are given four options (A, B, C, D) that discuss whether overdetermined systems can be consistent. Each option provides a specific system of three equations in two unknowns and claims whether the system is consistent or not.

Let's evaluate the systems provided in each option:

• Option A: The system is: $$x_{1}=2, \quad x_{2}=4, \quad x_{1}+x_{2}=24$$ Substituting $x_{1} = 2$ and $x_{2} = 4$ into the third equation gives $2 + 4 = 6$, which is not equal to 24. Therefore, this system is inconsistent.

• Option B: The system is: $$x_{1}=2, \quad x_{2}=4, \quad x_{1}+x_{2}=8$$ Substituting $x_{1} = 2$ and $x_{2} = 4$ into the third equation gives $2 + 4 = 6$, which is not equal to 8. Therefore, this system is inconsistent.

• Option C: The system is: $$x_{1}=2, \quad x_{2}=4, \quad x_{1}+x_{2}=6$$ Substituting $x_{1} = 2$ and $x_{2} = 4$ into the third equation gives $2 + 4 = 6$, which is equal to 6. Therefore, this system is consistent.

• Option D: The system is: $$x_{1}=2, \quad x_{2}=4, \quad x_{1}+x_{2}=12$$ Substituting $x_{1} = 2$ and $x_{2} = 4$ into the third equation gives $2 + 4 = 6$, which is not equal to 12. Therefore, this system is inconsistent.

Final Answer