INVESTIGATION: Is there always a solution?

Task:

You are given three systems of simultaneous linear equations:

System A: 

$\displaystyle \begin{cases} x+y&=4 \\ 2x-y&=1 \end{cases}$   

System B:     

 $\displaystyle \begin{cases} x+y&=3 \\ x+y&=5 \end{cases}$                  

System C:

$\displaystyle \begin{cases} 2x-3y&=6 \\ 4x-6y&=12 \end{cases}$

Part 1: Graphing the system

1. Plot each system of equations on the same set of Cartesian axes. Label each line clearly.
2. Describe what you observe about the points of intersection (or lack thereof) for each system.
3. Explain how the graphs help you determine whether a system has one solution, no solution, or infinitely many.

Part 2: Solving with matrices

1.  Write each system as a matrix equation in the form $\displaystyle AX=B$, where:

   • $\displaystyle A$ is the coefficient matrix

   • $\displaystyle X$ is the variable matrix

   • $\displaystyle B$ is the constant matrix

2. Is the determinant alone enough to determine whether a system has one, none, or infinite solutions? Explain why or why not.
3. Use the determinant, and if necessary additional information, to justify whether the system has one solution, no solution, or infinitely many.
4. Compare your conclusion with your graphs from Part 1.

5. For the system with a unique solution, solve using the inverse matrix method. Describe what the solution represents on the graph from Part 1.

Part 3: Design your own

1. Create three systems of linear equations:
   • One with a unique solution
   • One with no solution
   • One with infinitely many solutions
2. Write each system in matrix form and calculate the determinant of the coefficient matrix to verify your predictions.