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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.

User Icon for Bot GPT 3.5 Bot GPT 3.5 answered
7 months ago

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.