Question

Determine if the statement is true or false.

If a linear system has the same number of equations and variables, then it must have a unique solution. Explain
A)True
B)False

Answer 1

The answer is False. A linear system with the same number of equations and variables does not always have a unique solution.

Step-by-step explanation:

The answer to this question is False. While it is true that a linear system with the same number of equations and variables has a chance of having a unique solution, it is not always the case. Whether a linear system has a unique solution depends on the consistency of the equations and the independence of the equations. If the equations are consistent and independent, then the system will have a unique solution. However, if the equations are inconsistent or dependent, the system may have infinitely many solutions or no solution at all.

For example, consider the following system of equations:

2x + 3y = 5
4x + 6y = 10

These two equations are consistent and independent, and they represent intersecting lines on a graph. Therefore, the system has a unique solution (x = 1, y = 1).