How to Prove a Number Trick Always Works
Learn to simplify by collecting like terms, open brackets without losing a sign, test whether two expressions are equivalent, and use algebra to explain why a trick never fails.
How do you simplify an algebraic expression?
By collecting like terms — adding or subtracting the coefficients of terms that share exactly the same letters, and leaving unlike terms as they are.
This page covers everything in the CBSE Class 7 Mathematics chapter's second half: simplifying by collecting like terms, removing brackets, deciding whether two expressions are equivalent, and using algebra to explain why a number pattern or trick always works.
This page covers everything in the CBSE Class 7 Mathematics chapter's second half: simplifying by collecting like terms, removing brackets, deciding whether two expressions are equivalent, and using algebra to explain why a number pattern or trick always works.
How do you collect like terms correctly?
Group the terms that match, then combine only their coefficients. The letters themselves do not change.
Simplify :
The answer stops there. and are unlike, so they cannot be combined into a single term — and writing would be wrong.
Another: , remembering that the lone counts as .
For example, if a shop sells pens and notebooks on Monday and more on Tuesday, the weekly totals for pens and notebooks stay separate — you cannot add pens to notebooks, and that is exactly what unlike terms mean.
The sign must travel with its term. In the terms are , and , and dropping the minus turns the answer into .
Simplify :
The answer stops there. and are unlike, so they cannot be combined into a single term — and writing would be wrong.
Another: , remembering that the lone counts as .
For example, if a shop sells pens and notebooks on Monday and more on Tuesday, the weekly totals for pens and notebooks stay separate — you cannot add pens to notebooks, and that is exactly what unlike terms mean.
The sign must travel with its term. In the terms are , and , and dropping the minus turns the answer into .
Formula
How do you remove brackets safely?
The distributive property spreads a multiplier across everything inside a bracket:
So , and .
The sign in front of the bracket decides what happens when there is no multiplier:
- a plus leaves all signs unchanged:
- a minus reverses every sign inside:
A worked example combining both:
Notice gave . Missing that single sign is the most frequent error in the chapter.
Check by substituting. With : the original is , and the simplified form gives . They agree, so the simplification is right.
So , and .
The sign in front of the bracket decides what happens when there is no multiplier:
- a plus leaves all signs unchanged:
- a minus reverses every sign inside:
A worked example combining both:
Notice gave . Missing that single sign is the most frequent error in the chapter.
Check by substituting. With : the original is , and the simplified form gives . They agree, so the simplification is right.
How do you tell whether two expressions are equivalent?
Two expressions are equivalent if they give the same value for every value of the letter. There are two ways to check.
Simplify both and see whether they match. Is equivalent to ? Expanding the first gives , so yes.
Substitute values and compare. Is equivalent to ? Try : the first gives , the second gives . They differ, so they are not equivalent.
The two methods have different strengths. Substituting a value can disprove equivalence with a single counterexample — one mismatch settles it. But agreeing on one value does not prove equivalence, since two different expressions can coincide by accident at a particular number.
For example, and both give 4 when , yet they are plainly not equivalent — try and they give 9 and 6.
So to prove equivalence, simplify; to disprove it, one well-chosen substitution is enough.
Simplify both and see whether they match. Is equivalent to ? Expanding the first gives , so yes.
Substitute values and compare. Is equivalent to ? Try : the first gives , the second gives . They differ, so they are not equivalent.
The two methods have different strengths. Substituting a value can disprove equivalence with a single counterexample — one mismatch settles it. But agreeing on one value does not prove equivalence, since two different expressions can coincide by accident at a particular number.
For example, and both give 4 when , yet they are plainly not equivalent — try and they give 9 and 6.
So to prove equivalence, simplify; to disprove it, one well-chosen substitution is enough.
Why does a number trick always work?
Because algebra lets you follow any starting number at once, instead of testing examples one by one.
Take this trick: think of a number, add 5, double it, subtract 10, then halve it. You always get back your original number.
Let the number be and follow the steps:
The result is — whatever was. That is a proof, not a demonstration: no amount of trying numbers could establish it, but one line of algebra does.
The same tool states general formulas. The sum of three consecutive numbers is , which shows the total is always three times the middle number — a fact you could notice from examples but only explain with letters.
That is the real point of this chapter: letters do not just shorten arithmetic, they let you say something is true for every case.
Take this trick: think of a number, add 5, double it, subtract 10, then halve it. You always get back your original number.
Let the number be and follow the steps:
The result is — whatever was. That is a proof, not a demonstration: no amount of trying numbers could establish it, but one line of algebra does.
The same tool states general formulas. The sum of three consecutive numbers is , which shows the total is always three times the middle number — a fact you could notice from examples but only explain with letters.
That is the real point of this chapter: letters do not just shorten arithmetic, they let you say something is true for every case.
Exam tip
Exam tip: expanding brackets in two stages
Nearly every lost mark here comes from a sign dropped while opening a bracket, usually because the expansion and the collection were done in the same line.
Split it. First write the expression fully expanded, with every sign already flipped where needed:
Only then collect like terms to get .
Finish with a substitution check. Pick a small value such as , evaluate the original and your answer, and confirm they match. It costs two lines and catches a sign error that would otherwise cost the whole question — and in "show that" questions the check itself often earns credit.
Split it. First write the expression fully expanded, with every sign already flipped where needed:
Only then collect like terms to get .
Finish with a substitution check. Pick a small value such as , evaluate the original and your answer, and confirm they match. It costs two lines and catches a sign error that would otherwise cost the whole question — and in "show that" questions the check itself often earns credit.
Did you know
Why can't one matching value prove two expressions are equivalent?
Because two genuinely different expressions can cross paths at a particular number by coincidence.
and both equal 4 at , and and both equal 5 at . Testing only that value would suggest each pair is equivalent, and each pair is not.
That asymmetry is worth holding on to: a single counterexample disproves, but no number of agreements proves. For proof you must simplify.
and both equal 4 at , and and both equal 5 at . Testing only that value would suggest each pair is equivalent, and each pair is not.
That asymmetry is worth holding on to: a single counterexample disproves, but no number of agreements proves. For proof you must simplify.
Key takeaways
Simplifying expressions: quick revision
- Collect like terms by combining coefficients; unlike terms such as and stay separate, and every sign travels with its term.
- The distributive property gives ; a plus before a bracket keeps the signs and a minus reverses every one of them.
- Expand first, collect second — never both in one line — and check by substituting a small value.
- Two expressions are equivalent if they agree for every value: simplify to prove it, and one counterexample to disprove it.
- Following a number trick with a letter proves it works for every starting number, which examples alone never can.
You will remember all of this far better after answering five questions on it than after reading it twice.
- The distributive property gives ; a plus before a bracket keeps the signs and a minus reverses every one of them.
- Expand first, collect second — never both in one line — and check by substituting a small value.
- Two expressions are equivalent if they agree for every value: simplify to prove it, and one counterexample to disprove it.
- Following a number trick with a letter proves it works for every starting number, which examples alone never can.
You will remember all of this far better after answering five questions on it than after reading it twice.