To prove a linear system, you must show that all the equations in the system can be satisfied by the same set of values for the variables. This can be done by using various methods such as substitution, elimination, or graphing. If a solution exists for all equations in the system, then it is proven to be linear.
In a consistent linear system, there exists at least one solution that satisfies all the equations in the system. In an inconsistent system, there is no solution that satisfies all the equations. This means that the equations are contradictory and cannot be solved simultaneously.
Yes, a linear system can have one, infinite, or no solutions. If a system has one solution, it is called a consistent system with a unique solution. If it has infinite solutions, it is called a consistent system with infinitely many solutions. If there are no solutions, it is called an inconsistent system.
Matrices can be used to represent a linear system in a compact and organized manner. By performing row operations on the matrix, such as row reduction, you can determine the solution(s) to the system. If you end up with a reduced row echelon form of the matrix, you can easily read off the solution(s) to the system.
Proving a linear system is important because it allows us to determine whether the system has a solution or not. This can be useful in various fields such as engineering, physics, and economics where linear systems are commonly used to model real-world situations. Knowing the properties of a linear system also helps us in solving and analyzing more complex systems.