Understanding Consistency in Systems of Equations | The Key to Solving Math Problems

consistent

In mathematics, the term “consistent” is used in various contexts, but generally it refers to a system or set of equations that has at least one solution that satisfies all the given conditions

In mathematics, the term “consistent” is used in various contexts, but generally it refers to a system or set of equations that has at least one solution that satisfies all the given conditions. It implies that the equations do not contradict each other and can be satisfied simultaneously.

In the context of systems of linear equations, a consistent system has one or infinitely many solutions. This means that the equations in the system can be solved simultaneously, and there is either a unique solution or an infinite number of solutions that can satisfy all the equations.

For example, consider the following system of linear equations:
2x + 3y = 8 …(1)
4x – 3y = 10 …(2)

By solving this system, we can find that x = 2 and y = 2, which satisfies both equations. Therefore, the system is consistent with a unique solution.

In contrast, an inconsistent system of equations has no solution. This occurs when the equations in the system contradict each other and cannot be simultaneously satisfied.

For example, consider the following system of linear equations:
2x + 3y = 8 …(1)
4x – 3y = 5 …(2)

By attempting to solve this system, we find that adding equation (1) and equation (2) results in 6x = 13, which is not possible. This shows that the equations contradict each other, and the system is inconsistent.

In summary, a consistent system of equations has one or infinitely many solutions that can satisfy all the given equations, while an inconsistent system has no solution. The concept of consistency is fundamental in solving systems of equations and understanding the relationships between variables.

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