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

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
Transcript text: 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. \[ x_{1}=2, x_{2}=4, x_{1}+x_{2}=24 \] C. Yes, overdetermined systerns can be consistent. For example, the system of equations below is consistent because it has the solution (Type an ordered pair.) $\square$ \[ x_{1}=2, x_{2}=4, x_{1}+x_{2}=6 \] 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.) \[ x_{1}=2, x_{2}=4, x_{1}+x_{2}=8 \] D. No, overdetermined systems cannot be consistent because there are fewer free variables than equations. For example, the systern of equations below has no solution. \[ x_{1}=2, x_{2}=4, x_{1}+x_{2}=12 \]
failed

Solution

failed
failed

Solution Steps

Step 1: Understanding Overdetermined Systems

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.

Step 2: Analyzing the Given Options

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.

Step 3: Evaluating the Systems

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

The correct answer is C. Overdetermined systems can be consistent. For example, the system of equations below is consistent because it has the solution \((2, 4)\): \[ x_{1}=2, \quad x_{2}=4, \quad x_{1}+x_{2}=6 \] \[ \boxed{\text{The answer is C.}} \]

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Solve for (y). [ -4(-4 y+9)-y=7(y-6)-4 ] Simplify your answer as much as possible. [ y= ]
failedAnswered
(x)(3xy + 6x - 1)
failedAnswered
Find the derivative of the function. y=-4 e^(2 x^2)
failedAnswered
Question 3 3 pts What effect does judicial review have on the U.S. system of government? It makes the Supreme Court more powerful than Congress and the president. It allows the Supreme Court to overturn constitutional amendments. It allows the Supreme Court to perform a check on the constitutionality of government actions. It permits the Supreme Court to propose amendments to the Constitution.
failedAnswered
WORKERSCOMPCLCS Time Remaining 02:41:22 25 Alcohol or drug consumption at work: Review Later never can be used as a basis to deny workers compensation benefits to an injured employee. may be used as a basis to deny workers compensation benefits to an injured employee because the workers compensation policy excludes drug and alcohol-related injuries. can be used to deny workers compensation benefits if the employee using the alcohol or drugs is considered a substance abuser. may be used to deny workers compensation benefits to an injured worker if the workers was incapable of doing his job because of the alcohol or drugs. Page 25 of 50 Prev Page Next Page