To understand consistent systems of linear equations, let's break it down into simpler ideas:

Important Terms

1. Consistent System: This means the equations have at least one solution.
2. Inconsistent System: This means the equations have no solutions.
3. Dependent System: This means the equations are really just the same line, giving us endless solutions.

Types of Solutions

• Unique Solution: This is when two lines cross at just one point. This shows that the system is consistent.
• Infinitely Many Solutions: This happens when the equations show the same line (dependent). This also makes it a consistent system.
• No Solution: This occurs when the lines are parallel and never touch, which means the system is inconsistent.

How to Identify Consistent Systems

You can check if a system is consistent by doing these steps:

1. Graphing Equations: Draw the equations on the same graph.

   • If the lines cross, it’s a consistent system with a unique solution.
   • If the lines overlap perfectly, it’s a consistent system with infinitely many solutions.
   • If the lines are parallel and don’t meet, the system is inconsistent.

2. Finding the Determinant: For two equations that look like $Ax + By = C$, you can find the determinant: $D = A_1B_2 - A_2B_1$

   • If $D \neq 0$, it’s a consistent system with a unique solution.
   • If $D = 0$ but the lines aren’t the same, it’s an inconsistent system. If they are the same, it’s consistent.

Conclusion

In short, consistent systems can either have a unique solution or infinitely many solutions. You can find this out by either graphing the equations or checking the determinants. This knowledge is really helpful for solving and understanding linear equations in Grade 12 Algebra I.