Homogeneous and Nonhomogeneous Linear Systems of Equations

Is having a unique solution for the corresponding homogeneous system guarantee a unique solution for the nonhomogeneous system?

a. True b. False

Answer:

b. False

Having a unique solution for the corresponding homogeneous system does not guarantee a unique solution for the nonhomogeneous system. The statement is false.

In a nonhomogeneous system of linear equations, you have both the homogeneous part (where the right-hand side is all zeros) and a non-zero particular solution. The presence of the non-zero particular solution can introduce multiple solutions to the nonhomogeneous system even if the corresponding homogeneous system has a unique solution.

For example, consider the following nonhomogeneous system:

2x + 3y + z = 5

4x + 6y + 2z = 10

6x + 9y + 3z = 15

The corresponding homogeneous system is:

2x + 3y + z = 5

4x + 6y + 2z = 10

6x + 9y + 3z = 15

The homogeneous system has a unique solution, which is the trivial solution x = 0, y = 0, and z = 0. However, the nonhomogeneous system has an infinite number of solutions because you can add any particular solution to the homogeneous solution to get another solution. For example, if you take x = 1, y = 0 and z = 3, you get a solution to the nonhomogeneous system:

2(1) + 3(0) + 3 = 5

4(1) + 6(0) + 2(3) = 10

6(1) + 9(0) + 3(3) = 15

So, the nonhomogeneous system has multiple solutions, and it does not have a unique solution, even though the corresponding homogeneous system has a unique solution.