See the idea
Before you start
Understand that the equals sign means two expressions have the same value. Recall inverse operation pairs: addition/subtraction and multiplication/division.
Words used in this guide
- Variable
- A symbol representing a number, such as x.
- Solution
- A value that makes the original equation true.
- Inverse operation
- An operation that undoes another under the relevant conditions.
Why the method works
An equation is like a balanced comparison: both sides name the same amount. If you add the same number to both sides, the new values remain equal; the same is true for subtraction and for multiplying or dividing by a non-zero number. The goal is to leave the variable by itself. “Move it across and change the sign” can hide these reasons, so name the operation on both sides.
Undo addition with subtraction
For x + 7 = 12, subtract 7 from both sides. Then x = 5. Saying move the 7 can hide the reason the operation works.
Undo multiplication with division
For 4x = 28, divide both sides by 4 to get x = 7. Here 4x means four times x. Division by zero is not allowed.
Check in the original equation
Put the value back into the original statement. If x = 5, then x + 7 becomes 5 + 7 = 12. A check should use the original equation, not just the final line.
Worked examples, step by step
Example 1: Solve y − 6 = 9.
- Add 6 to both sides.
- y = 15. Check: 15 − 6 = 9.
Example 2: Solve x − 9 = 14.
- Undo subtracting 9 by adding 9 to both sides: x − 9 + 9 = 14 + 9.
- x = 23.
- Check in the original equation: 23 − 9 = 14.
Example 3: Solve y/5 = 7.
- Undo division by 5 by multiplying both sides by 5.
- y = 35.
- Check: 35/5 = 7. Dividing 7 by 5 would repeat the operation instead of undoing it.
Example 4: Four identical notebooks cost 18 currency units. What is the price p of one notebook?
- Equal prices give the equation 4p = 18. Divide both sides by 4.
- p = 4.5 currency units.
- Check: four notebooks at 4.5 each cost 18. A solution need not be a whole number.
Common mistakes and how to repair them
“Do the operation only on the side with x.”
If x + 4 = 11, subtracting 4 on the left alone changes the equality. Subtract 4 from 11 too, giving x = 7.
“4x means 4 + x.”
In algebra, 4x means 4 × x. The expressions 4 + x and 4x require different inverse operations.
Check your understanding
Choose an answer for each question. You can check again after reading the explanation.
Worked answers
Solve x + 8 = 13. 5. Subtract 8 from both sides; 5 + 8 = 13.
Solve 3x = 24. 8. Divide both sides by 3.
Practise independently
Work these out on paper. Include a diagram or a sentence where the question asks you to explain. Open each solution after making an attempt.
3. Solve a + 6 = 19.
Show the reasoning
a = 13. Subtract 6 from both sides; 13 + 6 = 19.
4. Solve 7b = 42.
Show the reasoning
b = 6. Divide both sides by 7; 7 × 6 = 42.
5. Solve c/4 = 2.5.
Show the reasoning
c = 10. Multiply both sides by 4; 10/4 = 2.5.
6. Solve d + 5 = 2.
Show the reasoning
d = −3. Subtract 5 from both sides; −3 + 5 = 2.
If a method still feels uncertain, revisit the closest worked example, then try a fresh problem. A correct answer copied from a solution tells you less than an explanation you can give independently.
Learn it → Try it → Explain it
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Questions students and parents ask
Can the variable be on the right?
Yes. 12 = x + 5 says the same thing as x + 5 = 12, so x = 7.
Why check in the original equation?
A mistaken rearrangement may look internally consistent. Substitution into the original statement checks whether the proposed answer actually solves the problem.
Can I divide both sides by zero?
No. Division by zero is undefined, so it is not a valid step.
Help at home: Talk through the balance
Use a covered cup for an unknown count and add three visible counters beside it. Say the total equals eight counters on the other side. Remove three from both sides.
Ask why both sides changed. If the child says “the 3 moved”, ask them to describe what happened to the counters.
Give an equation with a different inverse operation, such as x/3 = 4. Ask for a solution and an independent substitution check.
Keep the conversation focused on the reasoning you can hear or see. One short activity cannot establish lasting mastery or diagnose a learning difficulty.
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