Understanding Consistency in Linear Systems

The video tutorial explains how to determine the value of k for which a linear system is consistent. It begins by defining consistency in terms of solutions, then forms an augmented matrix from the given equations. Through a series of row operations, the matrix is simplified to reduced row echelon form. The tutorial then solves for k to ensure the system's consistency and verifies the solution graphically.

What does it mean for a linear system to be consistent?

• It has no solutions.

• It has a unique solution.

• It has one or infinite solutions.

• It has exactly two solutions.

In the context of the given system, what role does 'k' play?

• It is a variable in the first equation.

• It is a constant in the second equation.

• It is a coefficient in the third equation.

• It is a multiplier for the entire system.

What is the first step in transforming the augmented matrix?

• Making all diagonal elements equal to one.

• Adding a constant to all rows.

• Obtaining zeros in row two and row three, column one.

• Replacing row one with row two.

How do you make the entry in row two, column two a one?

• By adding a constant to row two.

• By multiplying row two by one twenty-second.

• By dividing row two by its first element.

• By subtracting row three from row two.

What is the purpose of obtaining a zero in row three, column two?

• To eliminate the need for further calculations.

• To ensure the system has no solutions.

• To simplify the matrix further.

• To make the matrix symmetric.

What is the final step to achieve reduced row echelon form?

• Multiplying row one by a constant.

• Adding row three to row one.

• Subtracting row two from row one.

• Replacing row one with a multiple of row two.

How is the value of 'k' determined for consistency?

• By setting the entire matrix to zero.

• By solving the equation negative eight k plus 106 equals zero.

• By making all elements in the matrix equal.

• By ensuring all rows are identical.

What does the solution 'k = 53/4' imply about the system?

• The system has no solutions.

• The system is inconsistent.

• The system is consistent with one solution.

• The system has multiple solutions.

How can the solution be verified graphically?

• By plotting the equations and checking for parallel lines.

• By checking if the lines overlap completely.

• By ensuring the lines do not intersect.

• By plotting the equations and finding the intersection point.

What is the significance of the intersection point in the graph?

• It indicates the system has no solutions.

• It represents the solution to the system.

• It shows the system is inconsistent.

• It confirms the system has infinite solutions.