Asked by UnicornSparkles

Which of the following is an example of an equation with no solution?(1 point)
Responses

A: 3x+4=3x+4

B: 3x+4=3x+3


C: 4x+3=3x+3

D: 3x+4=4x+3
Answered by UnicornSparkles
Which of the following is an example of an equation with exactly one solution?(1 point)
Responses

A: 3x+4=3x+3

B: 4x+3=4x+3

C: 3x+4=3x+4

D: 3x+4=4x+3
Answered by GPT-4o mini
An equation with no solution occurs when the variables cancel out and you are left with a false statement.

Let's go through the provided options:

A: \(3x + 4 = 3x + 4\)
This simplifies to a true statement (identity) since both sides are equal, so it has infinite solutions.

B: \(3x + 4 = 3x + 3\)
Subtracting \(3x\) from both sides gives \(4 = 3\), which is false. So, this equation has no solution.

C: \(4x + 3 = 3x + 3\)
Subtracting \(3x\) from both sides gives \(x + 3 = 3\), which has a solution.

D: \(3x + 4 = 4x + 3\)
Subtracting \(3x\) from both sides gives \(4 = x + 3\), which has a solution.

Therefore, the correct answer is:

**B: 3x + 4 = 3x + 3** (no solution)
Answered by GPT-4o mini
An equation with exactly one solution has a unique value for the variable when both sides are not identically equal or contradictory. Let's evaluate the options:

A: \(3x + 4 = 3x + 3\)
This simplifies to \(4 = 3\), which is a contradiction (false). Therefore, it has no solutions.

B: \(4x + 3 = 4x + 3\)
This simplifies to \(4x + 3 = 4x + 3\), which is an identity (true for all \(x\)). Thus, it has infinite solutions.

C: \(3x + 4 = 3x + 4\)
This is also an identity, so it has infinite solutions.

D: \(3x + 4 = 4x + 3\)
Subtracting \(3x\) from both sides gives \(4 = x + 3\). Rearranging gives \(x = 1\). This equation has exactly one solution.

Therefore, the correct answer is:

**D: 3x + 4 = 4x + 3** (exactly one solution)