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Question

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

3x+4=3x+4
3 x plus 4 equals 3 x plus 4

3x+4=3x+3
3 x plus 4 equals 3 x plus 3

3x+4=4x+3
3 x plus 4 equals 4 x plus 3

4x+3=3x+3

Answer 1

To determine which equation has no solution, we can analyze each option:

1. \(3x + 4 = 3x + 4\): This is true for all \(x\) (an identity), so it has infinitely many solutions.

2. \(3x + 4 = 3x + 3\): If we subtract \(3x\) from both sides, we get \(4 = 3\), which is false. Therefore, this equation has no solution.

3. \(3x + 4 = 4x + 3\): If we rearrange the equation, we can move \(3x\) to the right side to get \(4 = x + 3\), leading to \(x = 1\). So this equation has one solution.

4. \(4x + 3 = 3x + 3\): If we subtract \(3x\) from both sides, it reduces to \(x + 3 = 3\), leading to \(x = 0\). Therefore, this equation has one solution.

Based on this analysis, the equation with no solution is:

\(3x + 4 = 3x + 3\).

Answer 2

Which of the following is an example of an equation with exactly one solution?(1 point)
Responses

3x+4=4x+3
3 x plus 4 equals 4 x plus 3

3x+4=3x+3
3 x plus 4 equals 3 x plus 3

3x+4=3x+4
3 x plus 4 equals 3 x plus 4

4x+3=4x+3

Answer 3

To find the equation with exactly one solution, let's analyze each option:

1. \(3x + 4 = 4x + 3\): Rearranging gives \(4 - 3 = x\) or \(x = 1\). This equation has exactly one solution.

2. \(3x + 4 = 3x + 3\): This simplifies to \(4 = 3\), which is false. Thus, it has no solution.

3. \(3x + 4 = 3x + 4\): This is true for all \(x\), meaning it has infinitely many solutions.

4. \(4x + 3 = 4x + 3\): Similar to the previous one, this is true for all \(x\), leading to infinitely many solutions.

Based on this analysis, the equation that has exactly one solution is:

\(3x + 4 = 4x + 3\).

Answer 4

Solve for x to determine if the equation has one, none, or infinite solutions.
3(x+2)−6=3x (1 point)
Responses

The equation has infinite solutions.
The equation has infinite solutions.

The equation has one solution: x=−3.
The equation has one solution: x equals negative 3 .

The equation has no solution.
The equation has no solution.

The equation has one solution: x=3.

Answer 5

Let's solve the equation \(3(x+2) - 6 = 3x\).

1. Distribute on the left side: \[ 3x + 6 - 6 = 3x \] Simplifying gives: \[ 3x = 3x \]

2. This is an identity, meaning it is true for all values of \(x\). Therefore, the equation has infinite solutions.

Based on this, the correct response is:

The equation has infinite solutions.

Answer 6

Solve for x to determine if the equation has one, none, or infinite solutions.
4x=3(x+5)−15 (1 point)
Responses

The equation has no solution.
The equation has no solution.

The equation has one solution: x=0.
The equation has one solution: x equals 0 .

The equation has infinite solutions.
The equation has infinite solutions.

The equation has one solution: x=−10.

Answer 7

Let's solve the equation \(4x = 3(x + 5) - 15\).

1. Start by distributing on the right side: \[ 4x = 3x + 15 - 15 \]

2. Simplify the right side: \[ 4x = 3x + 0 \] which simplifies to: \[ 4x = 3x \]

3. Now, subtract \(3x\) from both sides: \[ 4x - 3x = 0 \] This simplifies to: \[ x = 0 \]

Since we found a specific value \(x = 0\), the equation has one solution.

Therefore, the correct response is:

The equation has one solution: \(x=0\).

Answer 8

Solve for x to determine if the equation has one, none, or infinite solutions.
5(2x+2)−7x=3x+6 (1 point)
Responses

The equation has no solution.
The equation has no solution.

The equation has one solution: x=6.
The equation has one solution: x equals 6 .

The equation has one solution: x=10.
The equation has one solution: x equals 10 .

The equation has infinite solutions.

Answer 9

Let's solve the equation \(5(2x + 2) - 7x = 3x + 6\).

1. Distribute on the left side: \[ 5 \cdot 2x + 5 \cdot 2 - 7x = 3x + 6 \] This simplifies to: \[ 10x + 10 - 7x = 3x + 6 \]

2. Combine like terms on the left side: \[ (10x - 7x) + 10 = 3x + 6 \] This simplifies to: \[ 3x + 10 = 3x + 6 \]

3. Now, subtract \(3x\) from both sides: \[ 10 = 6 \]

Since \(10\) does not equal \(6\), this is a false statement, indicating that there is no solution.

Therefore, the correct response is:

The equation has no solution.