How to Know when a System of Equations Has NO Solution, or Infinitely Many Solutions

Many students assume that all systems of equations have solutions. This article will use three examples of systems of equations to show that this assumption is incorrect.

Things You'll Need

• paper and
• pencil

Instructions

1. 

   Given the System of Equations, x + y = 5 and x - y = 1, we need to find the numbers x and y that simultaneously satisfy both equations. We can solve this system by several different methods, the method we will apply here is the Addition/Subtraction Method. Please click on the Image for a better understanding.

2. 

   Given the System of Equations, x + y = 5 and 2x + 2y = 10, we need to find the numbers x and y that simultaneously satisfy both equations. We can solve this system by several different methods, the method we will apply here is the Addition/Subtraction Method. Please click on the Image for a better understanding.

3. We can conclude from Step #2 that any System of Equation that has one equation that is the multiple of another equation that system of equation will have infinitely many solutions.

4. 

   Given the System of Equations, x + y = 5 and 2x + 2y = 11, we need to find the numbers x and y that simultaneously satisfy both equations. We can solve this system by several different methods, the method we will apply here is the Addition/Subtraction Method. Please click on the Image for a better understanding.