Solve the equation for x: 2x + b = c

Check this process:
Step 1: 2x + b - b = c - b
Step 2: 2x = c - b
Step 3: 2x - 2 = c - b - 2
Step 4: x = c/2 - b - 2
Check all of the choices that are true statements.
A. There is no error in the process
B. The process has an error from step 1 to step 2
C. The process has an error from step 2 to step 3
D. The process has an error from step 3 to step 4

1 answer

Let's evaluate each step of the process given for solving the equation \(2x + b = c\):

  1. Step 1: \(2x + b - b = c - b\)

    • This step is correct. When you subtract \(b\) from both sides, you maintain the equality.
  2. Step 2: \(2x = c - b\)

    • This step is also correct. The equation simplifies to \(2x = c - b\) after \(b\) is cancelled out.
  3. Step 3: \(2x - 2 = c - b - 2\)

    • This step contains an error. The correct move from \(2x = c - b\) should be to isolate \(x\); there is no justification for subtracting 2 from both sides here.
  4. Step 4: \(x = \frac{c}{2} - b - 2\)

    • Since the previous step was incorrect, this equation is also incorrect. The correct logic from \(2x = c - b\) is to divide both sides by 2 to get \(x = \frac{c - b}{2}\).

Now that we have analyzed each step, let's evaluate the choices:

  • A. There is no error in the process: False (there is an error).
  • B. The process has an error from step 1 to step 2: False (steps 1 and 2 are correct).
  • C. The process has an error from step 2 to step 3: True (this is where the first error occurs).
  • D. The process has an error from step 3 to step 4: True (because step 3 was incorrect, step 4 is also incorrect).

Thus, the true statements are: C and D.