Examine the six different equations that follow. Determine which pairs of equations
have one solution, no solution, and infinite solutions.
• 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.
Pairs of equations with one solution:
2x+3=2x+7
7x+3=7x+3
Pairs of equations with no solution:
4x+22=8x+10
5x+5=5x+4
Pairs of equations with infinite solutions:
3x+18=5x+8
3(4x+8)=4(3x+6)
Equations that have one solution have different coefficients or constant terms on each side of the equation. In the first pair (2x+3=2x+7), the constant terms are different on each side, resulting in one solution. In the second pair (7x+3=7x+3), the coefficients are the same on each side, but the constant terms are different, resulting in one solution.
Equations that have no solution have inconsistent coefficients or constant terms on each side of the equation. In the first pair (4x+22=8x+10), the coefficients are different on each side, resulting in no solution. In the second pair (5x+5=5x+4), the constant terms are different on each side, resulting in no solution.
Equations that have infinite solutions have consistent coefficients or constant terms on each side of the equation. In the first infinite solutions pair (3x+18=5x+8), the coefficients are the same on each side, resulting in infinite solutions. In the second infinite solutions pair (3(4x+8)=4(3x+6)), the equations can be simplified and the variables cancel out, resulting in infinite solutions.
summarize this into one paragraph: Equations that have one solution have different coefficients or constant terms on each side of the equation. In the first pair (2x+3=2x+7), the constant terms are different on each side, resulting in one solution. In the second pair (7x+3=7x+3), the coefficients are the same on each side, but the constant terms are different, resulting in one solution.
Equations that have no solution have inconsistent coefficients or constant terms on each side of the equation. In the first pair (4x+22=8x+10), the coefficients are different on each side, resulting in no solution. In the second pair (5x+5=5x+4), the constant terms are different on each side, resulting in no solution.
Equations that have infinite solutions have consistent coefficients or constant terms on each side of the equation. In the first infinite solutions pair (3x+18=5x+8), the coefficients are the same on each side, resulting in infinite solutions. In the second infinite solutions pair (3(4x+8)=4(3x+6)), the equations can be simplified and the variables cancel out, resulting in infinite solutions.