A Weighing Balance Can Solve Any Equation for You
Learn to turn puzzles and patterns into equations, solve them with the balance model and inverse operations, handle brackets and unknowns on both sides, and check every answer by substitution.
How does a weighing balance help you find an unknown?
Because an equation is a balance. The two sides weigh the same, so anything you do to one side must be done to the other, and the scales stay level.
That single rule replaces every trick and shortcut in this chapter. It covers everything in the CBSE Class 7 Mathematics chapter on equations: turning a situation into an equation, solving by the balance model and by inverse operations, handling brackets and unknowns on both sides, and forming equations from word problems.
That single rule replaces every trick and shortcut in this chapter. It covers everything in the CBSE Class 7 Mathematics chapter on equations: turning a situation into an equation, solving by the balance model and by inverse operations, handling brackets and unknowns on both sides, and forming equations from word problems.
How do you turn a puzzle or a pattern into an equation?
Name the unknown with a letter-number, then write in symbols exactly what the situation says.
Start with a matchstick pattern. One triangle takes 3 matchsticks; each extra triangle joined on takes 2 more. So triangles need
Check it: gives matchsticks, and gives . If a pattern used 21 matchsticks, then .
Now a word problem. A pen costs ₹8 more than a pencil. Let the pencil cost ₹, so the pen costs ₹. Two pens and three pencils cost ₹46 becomes
Two habits make this reliable. Write down what the letter stands for, including the unit — *let be the pencil's cost in rupees* — and translate the sentence phrase by phrase rather than all at once.
The distinction to keep clear: an expression like has no equals sign and cannot be solved, while an equation like states a balance and has a solution.
Start with a matchstick pattern. One triangle takes 3 matchsticks; each extra triangle joined on takes 2 more. So triangles need
Check it: gives matchsticks, and gives . If a pattern used 21 matchsticks, then .
Now a word problem. A pen costs ₹8 more than a pencil. Let the pencil cost ₹, so the pen costs ₹. Two pens and three pencils cost ₹46 becomes
Two habits make this reliable. Write down what the letter stands for, including the unit — *let be the pencil's cost in rupees* — and translate the sentence phrase by phrase rather than all at once.
The distinction to keep clear: an expression like has no equals sign and cannot be solved, while an equation like states a balance and has a solution.
How does the balance model solve an equation?
Picture the equals sign as the pivot of a level balance. Whatever you add to, subtract from, multiply or divide one side by, you do to the other side too — and it stays level.
Solve . Take 5 from both sides:
Solve . Divide both sides by 3:
Combine them for . Add 4 to both sides to get , then divide by 3 to get .
Always check by substitution, putting your answer back into the original equation: , which matches. A check that fails tells you to look again before the marks are lost.
Trial and error also works for simple equations — testing in gives 8, too small, so try 5 — but it becomes hopeless once answers are fractions or negatives. The balance method never depends on guessing well.
One restriction: you may divide both sides by any number except zero, since division by zero is undefined.
Solve . Take 5 from both sides:
Solve . Divide both sides by 3:
Combine them for . Add 4 to both sides to get , then divide by 3 to get .
Always check by substitution, putting your answer back into the original equation: , which matches. A check that fails tells you to look again before the marks are lost.
Trial and error also works for simple equations — testing in gives 8, too small, so try 5 — but it becomes hopeless once answers are fractions or negatives. The balance method never depends on guessing well.
One restriction: you may divide both sides by any number except zero, since division by zero is undefined.
How do you solve equations with brackets or the unknown on both sides?
Clear the brackets first, then collect all the unknowns on one side using the same balance rule.
With brackets, multiply the bracket out:
The multiplier must reach every term inside — writing is the standard slip and gives a different answer.
With the unknown on both sides, subtract the smaller unknown term from both sides:
Subtract from both sides to get , add 3 to get , then divide by 2:
Substituting back checks both sides at once: the left gives and the right gives . Equal, so the answer is right.
When both brackets and split unknowns appear, the order is fixed: expand, collect the unknowns, collect the numbers, divide. Following it means you never have to decide what to do next.
With brackets, multiply the bracket out:
The multiplier must reach every term inside — writing is the standard slip and gives a different answer.
With the unknown on both sides, subtract the smaller unknown term from both sides:
Subtract from both sides to get , add 3 to get , then divide by 2:
Substituting back checks both sides at once: the left gives and the right gives . Equal, so the answer is right.
When both brackets and split unknowns appear, the order is fixed: expand, collect the unknowns, collect the numbers, divide. Following it means you never have to decide what to do next.
How do you form and solve equations from word problems?
Name one unknown, express everything else in terms of it, then write the equation the final sentence gives you.
Finish the pen-and-pencil problem from earlier:
So a pencil costs ₹6 and a pen ₹14. Check against the original wording: , and the pen is indeed ₹8 more.
An age problem works the same way. Ravi is 4 years older than his sister, and their ages add to 28. Let the sister be years old, so Ravi is :
The sister is 12 and Ravi is 16.
Notice the pen problem had two unknown prices but needed only one letter, because the second was described in terms of the first. That is the key move when a problem seems to have two unknowns: use the relationship between them to remove one.
And always answer the question actually asked. Solving for is not the answer if the question wanted the cost of a pen.
Finish the pen-and-pencil problem from earlier:
So a pencil costs ₹6 and a pen ₹14. Check against the original wording: , and the pen is indeed ₹8 more.
An age problem works the same way. Ravi is 4 years older than his sister, and their ages add to 28. Let the sister be years old, so Ravi is :
The sister is 12 and Ravi is 16.
Notice the pen problem had two unknown prices but needed only one letter, because the second was described in terms of the first. That is the key move when a problem seems to have two unknowns: use the relationship between them to remove one.
And always answer the question actually asked. Solving for is not the answer if the question wanted the cost of a pen.
Exam tip
Exam tip: always substitute your answer back
Equations are the one topic where you can check your own answer completely, and it takes one line.
Put your value into the original equation — not your rearranged version, which may already contain the mistake — and confirm both sides come out equal. For with , both sides give 22.
Set the solution out one step per line, with the operation you applied to both sides written beside it. Method marks are awarded per step, so a wrong final answer with correct working still scores.
When expanding a bracket, multiply every term inside it, and when moving a term across the equals sign, change its sign.
Finally, state the answer in words with its unit: a pencil costs ₹6. A bare leaves the examiner to guess which quantity you found.
Put your value into the original equation — not your rearranged version, which may already contain the mistake — and confirm both sides come out equal. For with , both sides give 22.
Set the solution out one step per line, with the operation you applied to both sides written beside it. Method marks are awarded per step, so a wrong final answer with correct working still scores.
When expanding a bracket, multiply every term inside it, and when moving a term across the equals sign, change its sign.
Finally, state the answer in words with its unit: a pencil costs ₹6. A bare leaves the examiner to guess which quantity you found.
Did you know
Why can you do the same thing to both sides without breaking an equation?
Because the equals sign is a statement that two amounts are identical, and identical amounts stay identical if you treat them identically.
Two bags balancing on scales stay balanced if you add a 2 kg weight to each, or if you remove half the contents of both. The balance is upset only by doing something to one side alone.
That is why solving an equation is not really a set of rules to memorise. Every step is the same idea — keep the scales level — applied with a different operation.
Two bags balancing on scales stay balanced if you add a 2 kg weight to each, or if you remove half the contents of both. The balance is upset only by doing something to one side alone.
That is why solving an equation is not really a set of rules to memorise. Every step is the same idea — keep the scales level — applied with a different operation.
Key takeaways
Linear equations: quick revision
- Name the unknown with a letter, state what it stands for and its unit, and translate the situation phrase by phrase.
- An expression like cannot be solved; an equation like can.
- Do the same operation to both sides to keep the balance: gives , and gives .
- With brackets, expand first and multiply every term inside; with unknowns on both sides, collect them on one side.
- The order never changes: expand, collect unknowns, collect numbers, divide — and you may divide by anything except zero.
- When a problem has two unknowns, express one in terms of the other, then check your answer by substituting into the original equation.
You will remember all of this far better after answering five questions on it than after reading it twice.
- An expression like cannot be solved; an equation like can.
- Do the same operation to both sides to keep the balance: gives , and gives .
- With brackets, expand first and multiply every term inside; with unknowns on both sides, collect them on one side.
- The order never changes: expand, collect unknowns, collect numbers, divide — and you may divide by anything except zero.
- When a problem has two unknowns, express one in terms of the other, then check your answer by substituting into the original equation.
You will remember all of this far better after answering five questions on it than after reading it twice.