x=? · Equation
Isolating the unknown without breaking the balance.
A pan holding 3 apples and a 2g weight balances an 11g weight. How much does one apple weigh?
Explain why equivalent operations on both sides help find an unknown.
Start with equality, addition and subtraction.
'Algebra' comes from a word meaning to restore a scale's balance.
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Keep the balance rather than move a number
An unknown number plus 3 is 7. Naming that number x gives x+3=7. The letter is a place for the number we do not yet know.
Subtract 3 from the left and also from the right. Equal amounts remain equal after subtracting the same amount. Thus x+3−3=7−3, giving x=4.
Saying that 3 moves across and becomes negative is shorthand for those subtractions. A number does not change its nature by crossing an equals sign.
The same idea applies to multiplication: divide both sides of 2x=8 by 2 to obtain x=4. Division by zero, however, is undefined. Not every operation is available without conditions.
Finally, put the proposed solution into the original equation. Since 4+3=7, it answers the original question. Follow the relation being preserved, not just a memorized procedure.
Read these symbols together
- x
- A number whose value is unknown or is being sought.
- =
- The values on the two sides are equal. It is not a signal to start the next calculation.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — a pan with 3 apples and a 2g weight balances an 11g weight. How much does one apple weigh?
Operate the scale yourself
Do the same thing to both pans and the balance survives.
✓ still balanced
🍎🍎🍎 + 2g = 11g
Read why a sign seems to change
An equation is a question; solving is scale surgery.
3x+2=11 asks: 'for which x do the two sides agree?' Every line of a solution is just a record of applying the same operation to both pans.
The goal isn't computation — it's isolation.
There is exactly one strategy: peel away everything stuck to x until x stands alone. Choosing the order of peeling is the whole of solving.
Operations that preserve solutions
Write the scale moves in mathematical language and you get the properties of equality.
Properties of equality
Add or multiply both sides by the same number and equality survives — like adding the same weight to both pans. (Division only by nonzero.)
What 'moving across' really is
The +2 didn't 'cross over' as −2 — both sides had 2 subtracted. The identity of the sign-flipping magic.
What a solution means
A solution is a value that makes the equation true when substituted. Checking isn't optional — it's the definition of a solution.
Make one more distinction
'The sign flips when a term moves across' is a rule to memorize.
It's not a rule — it's a footprint. Subtract 3 from both sides and the +3 on the left vanishes while a −3 appears on the right. It looks like the term 'crossed over and flipped', but really the same operation hit both pans.
An equation and an identity are the same thing.
2+3=5 is always true — an identity. 3x+2=11 is true only when x=3 — a condition. That's an equation, and solving it means finding the moment it becomes true.
Where else can this idea help?
Finding the pre-sale price
'$35 after a 30% discount' — the original price solves 0.7x = 35. Everyday reverse calculations are all equations.
Scaling a recipe
Turning a 4-serving recipe into 6 servings means solving a proportion equation for each ingredient.
Engineering unknowns
'How thick must this column be to bear that load?' — write the condition as an equation, isolate the unknown. Same structure.
Solving conditions in code
'When does the balance hit zero?' — even in programming, the thinking is: set up the equation, then solve it.
Explain it in your own words
Compare with one possible explanation
Subtracting 3 from both sides of x+3=7 cancels +3 on the left and leaves −3 on the right.
Try another question
Q15x − 7 = 18. Find x with exactly two scale moves.Show answer ▾
Add 7 to both sides → 5x = 25. Divide both sides by 5 → x = 5. No 'moving across' rule needed — two scale moves finish it.
Q2Is 2x + 6 = 2(x + 3) an equation? Can you find x?Show answer ▾
Expand: 2x+6 = 2x+6 — true for every x. That's an identity: the solution isn't 'one value' but 'all of them'. The reverse also exists — x = x+1 has no solution. Equations don't always hand you exactly one answer.
Q3'Twice a number, minus 4, equals the number itself' — write the equation and solve it.Show answer ▾
2x − 4 = x. Subtract x from both sides: x − 4 = 0; add 4: x = 4. Translating words into an equation is half of solving.
Try answering yourself before revealing it — getting it wrong is where learning starts.
The idea to take with you
Solving an equation uses permitted equivalent operations to find its solutions.
What question comes next?
A reading path containing this lesson: Unknown numbers and balance · 2/3
Previous concept
Exponents
Having learned a language for writing numbers (powers), it's time for a language for handling numbers you don't know yet.
← Exponents labLeads to next
Functions
Where an equation finds 'the single moment things balance', a function sees 'the whole relationship across every input'. From a point to a relationship — the view widens.
Go to the Functions lab →Related lab
Percent
Finding a pre-sale price IS solving an equation — see 0.7x = 35 in the field.
Go to the Percent lab →