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Liza needs help to solve this linear equation.

4(x – 5) + 2x = 9x – 2(4 + 2x)

She has completed two steps, but now she’s stuck. Examine her work.

1.   4x - 20 + 2x = 9x - 8 - 4x

2.   6x - 20 = 5x - 8

What could be the next step? Select all that apply.
Add 8 to both sides.
Add 5x to both sides.
Subtract 6x from both sides.
Subtract 20 from both sides.
Divide both sides by –8.
Divide both sides by –20.

1 answer

Liza's last equation is

\[ 6x - 20 = 5x - 8. \]

To solve for \( x \), we can isolate \( x \) by performing the following steps:

  1. Add 20 to both sides: This will eliminate the -20 on the left side. \[ 6x - 20 + 20 = 5x - 8 + 20 \] This results in: \[ 6x = 5x + 12. \]

  2. Subtract 5x from both sides: This will isolate \( x \) on the left side. \[ 6x - 5x = 5x + 12 - 5x \] This results in: \[ x = 12. \]

Now, let's examine the options provided:

  • Add 8 to both sides.: This is not the best move in Liza's current equation and won't lead to solving for \( x \).
  • Add 5x to both sides.: This will help move all \( x \)-terms to one side, making it a valid step.
  • Subtract 6x from both sides.: This will eliminate \( x \) from the left side, but it is generally not the preferred method.
  • Subtract 20 from both sides.: This will not isolate \( x \) and is not useful given the current equation.
  • Divide both sides by -8.: This is not applicable at this stage since we don't have a factor of -8.
  • Divide both sides by -20.: This is also not applicable at this stage since we don't have a factor of -20.

The useful next steps are:

  • Add 5x to both sides.

Thus, the main next step Liza should take is to add \( 5x \) to both sides. After that, she can easily isolate \( x \).