Question

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=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 infinite solutions.

Answer 1

To solve the equation \( 5(2x+2)−7x=3x+6 \), we first simplify the left side.

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

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

3. Now, let's isolate the terms involving \( x \):
\[
3x + 10 - 3x = 3x + 6 - 3x
\]
This simplifies to:
\[
10 = 6
\]

Since \( 10 \) does not equal \( 6 \), we have reached a contradiction. This indicates that there are no values of \( x \) that can satisfy the equation.

Thus, the conclusion is:

**The equation has no solution.**