Question

Examine the six different equations that follow. Determine which pairs of equations have one solution, no solution, and infinite solutions.

• Pick the two equations that have infinite solutions.
• Pick the two equations that have no solution.
• Pick 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)

Answer 1

[AI-generated answer — Bot GPT 3.5]

Equations with Infinite Solutions:

1. 7x + 3 = 7x + 3
2. 3(4x + 8) = 4(3x + 6)

Equations with No Solution:
1. 2x + 3 = 2x + 7
2. 3x + 18 = 5x + 8

Equations with One Solution:
1. 5x + 5 = 5x + 4
2. 4x + 22 = 8x + 10

Answer 2

Equations with Infinite Solutions:

1. 7x + 3 = 7x + 3
2. 3(4x + 8) = 4(3x + 6)

Equations with No Solution:
1. 2x + 3 = 2x + 7
2. 3x + 18 = 5x + 8

Equations with One Solution:
1. 5x + 5 = 5x + 4
2. 4x + 22 = 8x + 10

What do you notice about equations that are in each of these categories? Use complete sentences to answer the questions that follow.

Answer 3

[AI-generated answer — Bot GPT 3.5]

Equations with Infinite Solutions have the same expression on both sides, resulting in all real numbers being valid solutions.

Equations with No Solutions have contradictory expressions on both sides, making it impossible for any value to satisfy the equation.

Equations with One Solution have equivalent expressions on both sides except for one particular value, which makes the equation true. It is this unique solution that distinguishes these equations from those with infinite solutions and no solution.

Answer 4

What does it mean to have infinite solutions?

Answer 5

[AI-generated answer — Bot GPT 3.5]

Having infinite solutions in an equation means that any real number can satisfy the equation. This occurs when both sides of the equation are identical, meaning that no matter what value is substituted for the variable, the equation remains true. In other words, there is no specific value that can be identified as the solution because all real numbers work. This situation often arises when the equation can be simplified or manipulated to show that the expressions on each side are equivalent.

Answer 6

What does it mean to have no solutions?

Answer 7

[AI-generated answer — Bot GPT 3.5]

Having no solutions in an equation means that there is no value that can be substituted for the variable that will make the equation true. This occurs when the expressions on the two sides of the equation are contradictory or cannot be made equivalent through simplification. In other words, no matter what value is chosen for the variable, the equation cannot be satisfied. This scenario usually arises when there is a logical inconsistency in the equation, making it impossible for any real number to be a solution.

Answer 8

What does it mean to have one solution?

Answer 9

[AI-generated answer — Bot GPT 3.5]

Having one solution in an equation means that there is a unique value for the variable that makes the equation true. This value is the only one that satisfies the equation, and no other value can work. Equations with one solution usually involve expressions on both sides that are equivalent except for one specific value of the variable. When this value is substituted into the equation, it balances both sides, resulting in equality. The presence of one solution implies that the equation has a well-defined and single answer that meets the requirements of the given problem.