Solutions of Systems of Equations
Graphs of systems of linear equations may be intersecting lines, parallel lines, or the same line. Systems of equations can be described by the number of solutions they have.
In general, a solution of a system in two variables is an ordered pair that makes BOTH equations true.
In other words, it is where the two graphs intersect, what they have in common. So if an ordered pair is a solution to one equation, but not the other, then it is NOT a solution to the system.
A consistent system is a system that has at least one solution.
An inconsistent system is a system that has no solution.
The equations of a system are dependent if ALL the solutions of one equation are also solutions of the other equation. In other words, they end up being the same line.
The equations of a system are independent if they do not share ALL solutions. They can have one point in common, just not all of them.
Solutions of Systems of Equations
Practice: Determine whether a system of equations has one solution, no solution, or infinitely many solutions by graphing in the following exercises.