Complete missing values in linear equations according to the number of solutions

The  exercise appears under the 8th grade (U.S.) Math Mission. This exercise helps to understand the difference between equations with one solution, no solutions and many solutions.

Types of Problems
There is one type of problems in this exercise:


 * 1) Find the value of $$P$$ and $$Q$$ that have an equation with zero, one, or infinitely many solutions: This problem presents an equation and the student is asked to determine what values determine many solutions the equation has.[[File:Complete missing values in linear equations according to the number of solutions.PNG|thumb|Find the value of $$P$$ and $$Q$$ that have an equation with zero, one, or infinitely many solutions]]

Strategies
Knowledge of solving equations and plugging in values exercise accurately and efficiently.
 * 1) Equations should be solved in the standard way. If $$x$$ can be isolated, there is one solution. If the $$x$$s are reduced completely and what remains is a true statement, then there are infinitely many solutions. If the $$x$$s reduce and all that remains is a false statement, then there are no solutions.

Real-life applications

 * 1) An ability to recognize identities and contradictions can help with logic and reasoning in debate and critical thinking.
 * 2) The SAT has problems of this type.