Describe the three types of solutions systems of equations have when graphed.
When graphing systems of equations, there are three possible outcomes: no solution, one solution, or infinitely many solutions. These outcomes depend on the relationship between the lines represented by the equations.
1. No Solution: A system of equations has no solution when the lines representing the equations are parallel, meaning they do not intersect. In this case, the lines never cross each other and there is no common point of intersection. If you graph the equations and find that the lines are parallel, you can conclude that the system has no solution.
2. One Solution: A system of equations has one solution when the lines representing the equations intersect at a single point. In this case, there is one unique solution that satisfies both equations simultaneously. If you graph the equations and find that the lines intersect at a single point, the system has one solution.
3. Infinitely Many Solutions: A system of equations has infinitely many solutions when the lines representing the equations are coincident, meaning they overlap each other. In this case, any point on the line is a solution to the system because all points on the line satisfy both equations. If you graph the equations and find that the lines overlap completely, you can conclude that the system has infinitely many solutions.
To determine the type of solution, you can graph the equations on a coordinate plane or use algebraic methods such as substitution or elimination to analyze the equations.