A child who has only ever solved equations meets 3x + 2 = 3x + 5 as a dead end and assumes they made a mistake. They did not: the x really does disappear, and what is left — 2 = 5 — is never true, so no number works. That is the answer, not a failed calculation, and it is the single most useful thing on this page.
Why this sheet works
The method is the same for all three and it is a smaller question than solving: collect the x terms and see what survives. If an x is left, one value works. If the x vanishes and a false statement is left, nothing works. If the x vanishes and a true statement is left, everything does. A child who asks that question can answer an equation they have never seen.
How this one works
One question from this sheet, worked through a step at a time.
17x − 37 = 17x − 33
17x − 37 = 17x − 33
17x − 37 = 17x − 33
x is gone
−37 = −33— never true
no solution
Do not start by hunting for x. Ask a smaller question first: what is left when the x terms are collected on one side?
Both sides have the SAME number of x, so taking it off both sides removes x completely. That is not a mistake — it is the answer arriving early.
What is left is −37 = −33, which is never true, whatever x is. So there is no solution.
