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?
A. 3(x+4) = x + 4
B. 3(x+4) = x + 12
C. 3(x+4) = 3x + 12
D. 3(x+4) = 3x + 4

Answer 1

To find which step in Max's solution could lead to a conclusion that the equation has no solution, we need to analyze each option to see if it leads to an inconsistency when solved.

A. **3(x+4) = x + 4**
\[
3x + 12 = x + 4 \\
3x - x + 12 = 4 \\
2x + 12 = 4 \\
2x = 4 - 12 \\
2x = -8 \\
x = -4
\]
This has a solution.

B. **3(x+4) = x + 12**
\[
3x + 12 = x + 12 \\
3x - x + 12 = 12 \\
2x + 12 = 12 \\
2x = 12 - 12 \\
2x = 0 \\
x = 0
\]
This has a solution.

C. **3(x+4) = 3x + 12**
\[
3x + 12 = 3x + 12
\]
This equation is an identity (true for all x), which means it has infinitely many solutions, not no solution.

D. **3(x+4) = 3x + 4**
\[
3x + 12 = 3x + 4 \\
3x - 3x + 12 = 4 \\
12 = 4
\]
This is a contradiction (false statement), which indicates that there is no solution.

The correct answer is **D**. This step leads Max to conclude that the equation has no solution.