25. Solutions of Linear Systems: Existence, Uniqueness, General Form

Try these first

1. 1.

   Define a linear system of equations.

   Hint

   Think about equations that share variables.

2. 2.

   What is the general form of a linear system in matrix notation?

   Hint

   Remember: A, x, b!

3. 3.

   Which of the following is NOT a type of solution for linear systems?

   • Inconsistent
   • Unique
   • Circular

   Hint

   Recall the classifications discussed in class.

4. 4.

   True or False: The unique solution is present when Rank(A) equals the number of variables.

   • True
   • False

   Hint

   Think about solutions and their associated conditions.

5. 5.

   Consider two linear equations: y = 2x + 3 and y = -x + 1. Determine the type of solution and justify your answer.

   Hint

   Graph the equations to visualize their intersection.

6. 6.

   Given three equations in three variables: x + y + z = 3, x + 2y + z = 5, and 2x + 3y + 3z = 8, analyze the system for dependency and determine the rank.

   Hint

   Try row-reducing the augmented matrix.