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Look at the equation below.

−10x+13+12x=5x−8

Which of the following steps could be used to transform the equation into the form x=a
, where a
is a number, to show that the equation has one solution?

Select TWO options that apply.

(2 points)
Responses

Subtract 10x from both sides of the equation and then simplify.
Subtract 10x from both sides of the equation and then simplify.

Add 5x to both sides of the equation and then simplify.
Add 5x to both sides of the equation and then simplify.

Add 8 to both sides of the equation and then simplify.
Add 8 to both sides of the equation and then simplify.

Combine like terms 10x and 12x to get 22x, and then subtract 22x from both sides of the equation.
Combine like terms 10x and 12x to get 22x, and then subtract 22x from both sides of the equation.

Subtract 13 from both sides of the equation and then simplify.
Subtract 13 from both sides of the equation and then simplify.

1 answer

To solve the equation

\[ -10x + 13 + 12x = 5x - 8, \]

we want to manipulate it to get it into the form \(x = a\).

Let's analyze the given options:

  1. Subtract 10x from both sides of the equation and then simplify.

    • This is a reasonable step because it allows us to combine the x terms.
    • This would result in: \[ (12x - 10x) + 13 = 5x - 8 \implies 2x + 13 = 5x - 8. \]

  2. Add 5x to both sides of the equation and then simplify.

    • This step could also work but is not the simplest path; it leads to adding into both sides.
    • If we do this, we would end up with: \[ -10x + 13 + 12x + 5x = -8 \implies 7x + 13 = -8. \]

  3. Add 8 to both sides of the equation and then simplify.

    • This could help isolate constants but doesn't directly lead to x.
    • This would give us \( -10x + 13 + 12x + 8 = 0 \), and it's not effective for isolating x directly.

  4. Combine like terms 10x and 12x to get 22x, and then subtract 22x from both sides of the equation.

    • This is incorrect since adding terms should not result in 22x (should be combined correctly, producing 2x).

  5. Subtract 13 from both sides of the equation and then simplify.

    • This does help isolate x terms from constants. After performing this step, you will have all x terms on one side: \[ -10x + 12x = 5x - 8 - 13 \implies 2x = 5x - 21. \]

From the analysis, the two options that effectively transform the original equation into isolating \(x\) are:

  1. Subtract 10x from both sides of the equation and then simplify.
  2. Subtract 13 from both sides of the equation and then simplify.

So, the correct choices are:

  • Subtract 10x from both sides of the equation and then simplify.
  • Subtract 13 from both sides of the equation and then simplify.