Investigation: Determining the Number of Solutions to a System Algebraically

Glossary Checklist FAQ

Determining the Number of Solutions to a System Algebraically

1. A system of linear-quadratic equations consists of \(y = x^2 + 2\) and \(y = 0.5x + 1\). By graphing the system, it can be seen that there is no real solution to the system.
   
   
   
1. Attempt to solve the system of equations using elimination or substitution.
   
   
2. Predict the result you will see when attempting to algebraically solve any system without a real solution.
   
   
2. A system of quadratic-quadratic equations consists of \(y = 3(x - 1)^2 + 2\) and \(y = 3x^2 - 6x + 5\). By graphing the system, it can be seen that this system has an infinite number of real solutions.
   
   
   
   
1. Attempt to solve the system of equations using elimination or substitution.
   
   
2. Predict the result you will see when algebraically attempting to solve any system with an infinite number of real solutions.
   
   

The result of trying to solve a system of equations with no real solution and the result of trying to solve a system with an infinite number of real solutions are predictable. By recognizing this, the nature of the solutions can be determined from the solution outcome.