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

Responses

a) 3x+4=3x+3
3 x plus 4 equals 3 x plus 3

b) 3x+4=4x+3
3 x plus 4 equals 4 x plus 3

c) 4x+3=3x+3
4 x plus 3 equals 3 x plus 3

d) 3x+4=3x+4

11 answers

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

a) \(3x + 4 = 3x + 3\)
Subtract \(3x\) from both sides:
\(4 = 3\) (false statement, hence no solution)

b) \(3x + 4 = 4x + 3\)
Subtract \(3x\) from both sides:
\(4 = x + 3\)
Solve for \(x\):
\(x = 1\) (has a solution)

c) \(4x + 3 = 3x + 3\)
Subtract \(3x\) from both sides:
\(x + 3 = 3\)
Solve for \(x\):
\(x = 0\) (has a solution)

d) \(3x + 4 = 3x + 4\)
This is an identity (true for all \(x\), hence infinite solutions)

Based on this analysis, option a) \(3x+4=3x+3\) is an example of an equation with no solution.

Which of the following is an example of an equation with exactly one 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

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

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

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

  1. \(3x + 4 = 3x + 4\)
    This equation is true for all \(x\) (identity), resulting in infinite solutions.

  2. \(3x + 4 = 3x + 3\)
    Subtract \(3x\) from both sides:
    \(4 = 3\), which is a false statement, leading to no solution.

  3. \(4x + 3 = 4x + 3\)
    This is also an identity, true for all \(x\), resulting in infinite solutions.

  4. \(3x + 4 = 4x + 3\)
    Subtract \(3x\) from both sides:
    \(4 = x + 3\)
    Solve for \(x\):
    \(x = 1\), which means there is exactly one solution.

Based on this analysis, the equation with exactly one solution is option 4) \(3x + 4 = 4x + 3\).

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 no solution.
The equation has no solution.

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

The equation has one solution: x=3.

Let's solve the equation \(3(x+2)−6=3x\) step-by-step:

  1. Start with the equation: \[ 3(x + 2) - 6 = 3x \]

  2. Distribute \(3\) on the left side: \[ 3x + 6 - 6 = 3x \]

  3. Simplify the left side: \[ 3x = 3x \]

At this point, we have \(3x = 3x\), which is a true statement for all values of \(x\).

Since the left side and the right side are equal for all \(x\), the equation has infinite solutions.

Therefore, the correct response is: The equation has infinite solutions.

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

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

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

The equation has one solution: x=−10.

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

  1. Start with the equation: \[ 4x = 3(x + 5) - 15 \]

  2. Expand the right side: \[ 4x = 3x + 15 - 15 \]

  3. Simplify the right side: \[ 4x = 3x + 0 \] \[ 4x = 3x \]

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

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

Thus, the correct response is: The equation has one solution: x=0.

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 infinite solutions.
The equation has infinite solutions.

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

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

The equation has no solution.

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

  1. Start with the equation: \[ 5(2x + 2) - 7x = 3x + 6 \]

  2. Distribute \(5\) on the left side: \[ 10x + 10 - 7x = 3x + 6 \]

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

  4. Subtract \(3x\) from both sides: \[ 10 = 6 \]

This is a false statement (10 does not equal 6), which means there are no solutions to this equation.

Thus, the correct response is: The equation has no solution.

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