Question

One of Max's homework problems is to solve a linear equation. He solves the equation and finds that it has no solution.

Which of the following could be a step in Max's solution?

Responses

3(x + 4) = x + 4
3(x + 4) = x + 4

3(x + 4) = x + 12
3(x + 4) = x + 12

3(x + 4) = 3x + 12
3(x + 4) = 3x + 12

3(x + 4) = 3x + 4

Answer 1

To determine which of the steps could lead to the conclusion that the equation has no solution, we need to see if any of the steps results in a contradictory statement (such as a false statement like 0 = 1).

Let's examine each option:

1. **3(x + 4) = x + 4**

Distributing on the left side gives:
\(3x + 12 = x + 4\)
Rearranging leads us to:
\(3x - x = 4 - 12\)
\(2x = -8\)
\(x = -4\) (this has a solution, so this step does not indicate no solution)

2. **3(x + 4) = x + 12**

Distributing on the left gives:
\(3x + 12 = x + 12\)
Rearranging leads us to:
\(3x - x = 12 - 12\)
\(2x = 0\)
\(x = 0\) (this has a solution, so this step does not lead to no solution)

3. **3(x + 4) = 3x + 12**

Distributing on the left gives:
\(3x + 12 = 3x + 12\)
This simplifies to an identity, which is true for all values of \(x\). (indicates infinitely many solutions, thus does not indicate no solution)

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

Distributing on the left gives:
\(3x + 12 = 3x + 4\)
Rearranging leads us to:
\(3x - 3x = 4 - 12\)
\(0 = -8\) (this is a contradiction, which means there is no solution)

Based on the analysis, the step that could indicate that the equation has no solution is:

**3(x + 4) = 3x + 4**