Determine which pairs of equations have one solution no solution infinite solutions 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). Now what do you notice about equations that are in each of these categories?

Answer 1

- The pair of equations 2x+3=2x+7 and 7x+3=7x+3 both have no solution because in the first equation, the constant terms on both sides are not equal and in the second equation, the variables and constant terms are already equal on both sides.

- The pair of equations 4x+22=8x+10 and 5x+5=5x+4 both have one solution because in the first equation, when simplified, it becomes 4x - 8x = 10 - 22, which results in -4x = -12 and x = 3, and in the second equation, when simplified, it becomes 5x - 5x = 4 - 5, which results in 0 = -1, which is not true.

- The pair of equations 3x+18=5x+8 and 3(4x+8)=4(3x+6) both have infinite solutions because in the first equation, when simplified, it becomes 18 = 2x - x + 8, which results in 18 = x + 8, and in the second equation, when simplified, it becomes 12x + 24 = 12x + 24, which is already a true statement.

What is noticed about equations that are in each of these categories is:
- Equations with no solution have constant terms on both sides that are not equal.
- Equations with one solution have variables and constant terms on both sides that cancel out when simplified.
- Equations with infinite solutions result in a true statement when simplified.