Equivalent expressions, step by step
Two expressions are equivalent when they name the same number whatever you put in for x.
Which expression is equivalent to 3x + 8 + x?
Which is the same as 3x + 8 + x?
- Equivalent does not mean it looks similar — it means the two expressions give the SAME answer whichever number x turns out to be. That is a strong claim, and it is checkable.
- Only the x terms can join: 3x and the bare x are 4 lots of x. The 8 is not an x, so it cannot go in with them and stays where it is.
- Sweeping the 8 in as well would give 12x, and that is the commonest wrong answer here — 8 is 8, whatever x turns out to be.
- Check it by putting a number in — the same number in both. With x = 3: 3x + 8 + x is 20, and 4x + 8 is 20. Equal, so the two agree at least here.
- That is how to test any of the choices: put the same x into each, and only the equivalent one comes out equal. Pick something other than 0 and 1, though — at 1 every power of x is 1, so 3x and 3x² both come out 3; and at 0 every x term vanishes, so two expressions that differ only in their x part look identical. That is why the check above used 3.
How it works.
- 01
Equivalent does not mean it looks similar — it means the two expressions give the SAME answer whichever number x turns out to be. That is a strong claim, and it is checkable.
- 02
Only the x terms can join: 3x and the bare x are 4 lots of x. The 8 is not an x, so it cannot go in with them and stays where it is.
- 03
Sweeping the 8 in as well would give 12x, and that is the commonest wrong answer here — 8 is 8, whatever x turns out to be.
- 04
Check it by putting a number in — the same number in both. With x = 3: 3x + 8 + x is 20, and 4x + 8 is 20. Equal, so the two agree at least here.
- 05
That is how to test any of the choices: put the same x into each, and only the equivalent one comes out equal. Pick something other than 0 and 1, though — at 1 every power of x is 1, so 3x and 3x² both come out 3; and at 0 every x term vanishes, so two expressions that differ only in their x part look identical. That is why the check above used 3.