Examine the six different equations that follow. Determine which pairs of equations have one solution, no solution, and infinite solutions. On your worksheet,
put a circle around the two equations that have infinite solutions.
put a square around the two equations that have no solution.
underline the two equations that have one solution.
2x+3+2x+7
7x+3+7x+3
4x+22=8x+10
5x+5=5x+4
3x+18=5x+8
3(4x+8)=4(3x+6)
What do you notice about equations that are in each of these categories? Use complete sentences to answer the questions that follow.
Equations with one solution typically involve a unique set of values for the variable that satisfy the equation. Equations with no solution often result from contradictions, such as 0=1. Equations with infinite solutions usually come from identities, where the left side of the equation is equivalent to the right side for all real numbers. In this case:
One solution:
4x+22=8x+10
5x+5=5x+4
No solution:
2x+3+2x+7
7x+3+7x+3
Infinite solutions:
3x+18=5x+8
3(4x+8)=4(3x+6)
Overall, equations with one solution result in a unique value for the variable, equations with no solution lead to a contradiction, and equations with infinite solutions are typically identities that hold true for all possible values of the variable.