Every Equation Has a Solution, Except When It Does Not
Learn to move the variable to one side, clear brackets and fractions using the LCM, solve proportions by cross multiplication, and always verify your answer.
What does it mean to solve an equation?
It means finding the one value of the variable that makes the two sides genuinely equal.
An equation is a claim of balance. In the claim is that those two expressions have the same value — and that claim is true for exactly one value of .
Solving means finding it. Every step you take must keep the balance: whatever you do to the left side, you do to the right side too. Add the same thing to both sides, subtract the same thing from both, multiply or divide both by the same non-zero number, and the equality survives.
That single rule is the whole method. Everything in this page — moving terms, clearing brackets, wiping out denominators — is that rule applied to a different-looking equation. This page covers the first part of the ICSE Class 8 Mathematics chapter on linear equations in one variable.
An equation is a claim of balance. In the claim is that those two expressions have the same value — and that claim is true for exactly one value of .
Solving means finding it. Every step you take must keep the balance: whatever you do to the left side, you do to the right side too. Add the same thing to both sides, subtract the same thing from both, multiply or divide both by the same non-zero number, and the equality survives.
That single rule is the whole method. Everything in this page — moving terms, clearing brackets, wiping out denominators — is that rule applied to a different-looking equation. This page covers the first part of the ICSE Class 8 Mathematics chapter on linear equations in one variable.
How do you solve an equation with the variable on both sides?
Collect the variable terms on one side and the numbers on the other, then divide.
Worked example 1. Solve .
Subtract from both sides, and add to both:
Verify. Left side: . Right side: . Equal, so is correct.
Worked example 2. Solve .
Verify. and . Correct.
Worked example 3 — a negative answer. Solve .
Here the larger variable term is on the right, so collect there instead:
Verify. Left: . Right: . Correct.
Which side to collect on. Move the variable to whichever side has the larger coefficient, so the coefficient you end up dividing by is positive. Collecting on the left in the last example would have given , which is the same answer but one more sign to get wrong.
What a term does when it crosses the equals sign. It changes sign. That is not a separate rule to memorise — it is what subtracting from both sides looks like once written quickly. Understanding it as a subtraction is what stops you from changing the sign of a term that never moved.
Worked example 1. Solve .
Subtract from both sides, and add to both:
Verify. Left side: . Right side: . Equal, so is correct.
Worked example 2. Solve .
Verify. and . Correct.
Worked example 3 — a negative answer. Solve .
Here the larger variable term is on the right, so collect there instead:
Verify. Left: . Right: . Correct.
Which side to collect on. Move the variable to whichever side has the larger coefficient, so the coefficient you end up dividing by is positive. Collecting on the left in the last example would have given , which is the same answer but one more sign to get wrong.
What a term does when it crosses the equals sign. It changes sign. That is not a separate rule to memorise — it is what subtracting from both sides looks like once written quickly. Understanding it as a subtraction is what stops you from changing the sign of a term that never moved.
How do you clear brackets and fractions from an equation?
Expand every bracket first, then multiply the whole equation by the LCM of the denominators.
Worked example 1 — brackets on one side. Solve .
Note that . Collecting:
Verify. . Correct.
Worked example 2 — brackets on both sides. Solve .
Verify. and . Correct.
Worked example 3 — simple fractions. Solve .
The LCM of and is . Multiply every term by :
Verify. . Correct.
Worked example 4 — fractions with expressions on top. Solve .
The LCM of and is :
Verify. , and . Then . Correct.
The two habits that matter here. First, keep the numerator in a bracket when you multiply: is , not . Losing that bracket after a minus sign is the single most expensive mistake in this topic. Second, multiply the constant term too — the on the right became , and forgetting it ruins an otherwise perfect solution.
A fractional answer is a correct answer. Nothing says must be a whole number, and rounding to makes the equation false.
Worked example 1 — brackets on one side. Solve .
Note that . Collecting:
Verify. . Correct.
Worked example 2 — brackets on both sides. Solve .
Verify. and . Correct.
Worked example 3 — simple fractions. Solve .
The LCM of and is . Multiply every term by :
Verify. . Correct.
Worked example 4 — fractions with expressions on top. Solve .
The LCM of and is :
Verify. , and . Then . Correct.
The two habits that matter here. First, keep the numerator in a bracket when you multiply: is , not . Losing that bracket after a minus sign is the single most expensive mistake in this topic. Second, multiply the constant term too — the on the right became , and forgetting it ruins an otherwise perfect solution.
A fractional answer is a correct answer. Nothing says must be a whole number, and rounding to makes the equation false.
How do you solve an equation written as a proportion?
Cross multiply — multiply each numerator by the opposite denominator — and the fractions disappear.
If then . This is just multiplying both sides by , so it is the balance rule again rather than a new trick.
Worked example 1. Solve .
Verify. Left: . Right: . Correct.
Worked example 2 — the variable in a denominator. Solve .
Verify. . Cross-checking, and , so the fraction really is . Correct.
The check that this equation type demands. The denominator must not be zero, so would have to be rejected. Here , which is safe. Always confirm your answer does not make a denominator zero — an equation like looks as though works, and it does not, because division by zero is undefined.
Worked example 3. Solve .
Verify. . Correct.
When there is no solution, and when there are infinitely many. Solve . Subtracting leaves , which is false whatever is — so the equation has no solution. Now solve . Expanding gives , true for every value of — an identity, not an equation to solve.
Both outcomes announce themselves the same way: the variable vanishes. What is left over tells you which case you are in, and neither is a mistake in your working.
If then . This is just multiplying both sides by , so it is the balance rule again rather than a new trick.
Worked example 1. Solve .
Verify. Left: . Right: . Correct.
Worked example 2 — the variable in a denominator. Solve .
Verify. . Cross-checking, and , so the fraction really is . Correct.
The check that this equation type demands. The denominator must not be zero, so would have to be rejected. Here , which is safe. Always confirm your answer does not make a denominator zero — an equation like looks as though works, and it does not, because division by zero is undefined.
Worked example 3. Solve .
Verify. . Correct.
When there is no solution, and when there are infinitely many. Solve . Subtracting leaves , which is false whatever is — so the equation has no solution. Now solve . Expanding gives , true for every value of — an identity, not an equation to solve.
Both outcomes announce themselves the same way: the variable vanishes. What is left over tells you which case you are in, and neither is a mistake in your working.
Exam tip
Exam tip: verify every answer, it costs two lines
Substitute your answer back into the original equation — not into your own rearranged version, which may already carry an error. Two lines of arithmetic turn a hopeful answer into a certain one.
Collect the variable on the side with the larger coefficient so you divide by a positive number.
A term that crosses the equals sign changes sign, because crossing is subtraction written quickly.
Expand brackets before clearing fractions, and when you multiply through by the LCM, keep every numerator in a bracket: , never .
Multiply the constant term by the LCM too. The lone on the right becomes .
When multiplying out , remember both signs change: it is .
A fraction or a negative number is a valid answer. Do not round to .
For a proportion, cross multiply, then check that your answer does not make any denominator zero.
And if the variable disappears entirely, read what is left: a false statement means no solution, a true one means every value works.
Collect the variable on the side with the larger coefficient so you divide by a positive number.
A term that crosses the equals sign changes sign, because crossing is subtraction written quickly.
Expand brackets before clearing fractions, and when you multiply through by the LCM, keep every numerator in a bracket: , never .
Multiply the constant term by the LCM too. The lone on the right becomes .
When multiplying out , remember both signs change: it is .
A fraction or a negative number is a valid answer. Do not round to .
For a proportion, cross multiply, then check that your answer does not make any denominator zero.
And if the variable disappears entirely, read what is left: a false statement means no solution, a true one means every value works.
Did you know
Why the balance picture is worth keeping in your head
Think of an equation as a pair of scales that is exactly level. On the left pan sits ; on the right, . They balance.
Now remove from both pans. The scales stay level, because you took the same weight from each side. Add to both pans — still level. What remains is balancing , and dividing both pans into three equal parts gives one balancing .
The picture explains the one rule that is easy to break. If you subtract from the left and forget the right, the scales tip and every line after that describes a different problem. That is why the answer to why did my solution come out wrong is nearly always something was done to one side only.
It also explains why multiplying both sides by zero is forbidden. Empty both pans and they balance perfectly — but they would have balanced whatever was in them, so you have destroyed all the information the equation held. Dividing by an expression that might be zero has the same defect, which is exactly why the proportion examples above needed a denominator check.
The balance is not a childish picture to be outgrown. It is the reason the rules are what they are.
Now remove from both pans. The scales stay level, because you took the same weight from each side. Add to both pans — still level. What remains is balancing , and dividing both pans into three equal parts gives one balancing .
The picture explains the one rule that is easy to break. If you subtract from the left and forget the right, the scales tip and every line after that describes a different problem. That is why the answer to why did my solution come out wrong is nearly always something was done to one side only.
It also explains why multiplying both sides by zero is forbidden. Empty both pans and they balance perfectly — but they would have balanced whatever was in them, so you have destroyed all the information the equation held. Dividing by an expression that might be zero has the same defect, which is exactly why the proportion examples above needed a denominator check.
The balance is not a childish picture to be outgrown. It is the reason the rules are what they are.
Key takeaways
Solving linear equations: quick revision
- An equation claims two expressions are equal. Keep the balance: do the same thing to both sides.
- Variable on both sides: collect variable terms one side, numbers the other. gives , so ; both sides check as .
- gives , so and both sides are .
- Collect on the side with the larger coefficient: gives , so and both sides are .
- A term changes sign when it crosses the equals sign, because crossing is subtraction.
- Brackets first: becomes , so . Note .
- gives , so and both sides are .
- Clear fractions with the LCM: times gives , so .
- times gives , so and . Keep each numerator in a bracket, and multiply the constant too.
- Cross multiplication: gives . So gives and , with both sides .
- gives , so ; and gives , so .
- Check no denominator becomes zero — a value like in must be rejected.
- If the variable vanishes: leaves , so there is no solution; is true for all .
- Fractions and negatives are valid answers. Never round them.
Take five equations, solve them, then verify each by substitution before looking at any answer key — the habit of proving your own answer is worth more marks than any shortcut.
- Variable on both sides: collect variable terms one side, numbers the other. gives , so ; both sides check as .
- gives , so and both sides are .
- Collect on the side with the larger coefficient: gives , so and both sides are .
- A term changes sign when it crosses the equals sign, because crossing is subtraction.
- Brackets first: becomes , so . Note .
- gives , so and both sides are .
- Clear fractions with the LCM: times gives , so .
- times gives , so and . Keep each numerator in a bracket, and multiply the constant too.
- Cross multiplication: gives . So gives and , with both sides .
- gives , so ; and gives , so .
- Check no denominator becomes zero — a value like in must be rejected.
- If the variable vanishes: leaves , so there is no solution; is true for all .
- Fractions and negatives are valid answers. Never round them.
Take five equations, solve them, then verify each by substitution before looking at any answer key — the habit of proving your own answer is worth more marks than any shortcut.