Solve for x to determine if the equation has one, none, or infinite solutions.

Question

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=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 no solution.

Answer 1

To determine how many solutions the equation \( 5(2x+2) - 7x = 3x + 6 \) has, let's solve it step by step.

1. Distribute the \( 5 \) on the left side: \[ 5(2x) + 5(2) - 7x = 3x + 6 \] 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: \[ 10 = 6 \] This is a false statement.

Since we reached a contradiction, this indicates that there are no values of \( x \) that satisfy the equation.

Therefore, the equation has no solution.