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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.

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.

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.

1 answer

To solve the equation \(-10x + 13 + 12x = 5x - 8\) and get it into the form \(x = a\), we can follow certain steps to simplify and isolate \(x\).

Let's analyze each of the options offered:

  1. Subtract 10x from both sides of the equation and then simplify.
    Performing this step:
    \(-10x + 12x + 13 = 5x - 8\)
    simplifies to:
    \(2x + 13 = 5x - 8\)
    This step is valid for continuing to isolate \(x\).

  2. Combine like terms 10x and 12x to get 22x, and then subtract 22x from both sides of the equation.
    This is incorrect because we're combining terms incorrectly. The correct combination is \(-10x + 12x = 2x\), not \(22x\). This option does not lead to the desired transformation.

  3. Subtract 13 from both sides of the equation and then simplify.
    This step is indeed useful. After subtracting 13, we would have:
    \(-10x + 12x = 5x - 8 - 13\)
    which simplifies to:
    \(2x = 5x - 21\)
    It is a valid step towards transforming the equation into the desired form.

  4. Add 5x to both sides of the equation and then simplify.
    Adding \(5x\) to both sides:
    \(-10x + 12x + 5x + 13 = -8\) simplifies to:
    \(7x + 13 = -8\)
    This step is valid and will lead us towards isolating \(x\).

  5. Add 8 to both sides of the equation and then simplify.
    This can be performed as well, but it has less direct application to isolating \(x\) at this point in the equation. It can certainly be part of the process, but it's not one of the primary steps for isolating \(x\) directly from both sides of the equation.

Thus, the two most effective and useful options from the selection provided 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.

These steps will effectively lead towards getting the equation into the form \(x = a\).