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Number of Solutions for a Linear System of Equations

When a linear system of equations has two equations and two variables, there is the possibility of having either one solution, no solutions, or infinitely many solutions.

One Solution

If the equations represent two straight lines which are not parallell, meaning they have different slopes, then there is exactly one solution for the system. This is the point where the lines intersect.

No Solutions

If the lines in the equation system are parallel, meaning that they have the same slope but different y-intecepts, they will never intersect. This means that the system of equations has "no solution".

Infinite Number of Solutions

If the equations have the same slope and y-intercept, this means that they describe two copies of the same line. The equation system has an infinite number of solutions since the lines represent the same points.