'Algebra' comes from a word meaning to restore a scale's balance.
A pan holding 3 apples and a 2g weight balances an 11g weight. How much does one apple weigh?
Isolating the unknown without breaking the balance.
Experiment
Hands-on experiment
Predict first โ 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.
โ still balanced
๐๐๐ + 2g = 11g
Read more โ why it exists ยท insights ยท common mistakes ยท formulasExpand โพ
Why
Why does this exist?
'I multiplied a number by 3, added 2, and got 11. What was the number?' โ humanity has solved backwards questions like this since Babylonian clay tablets, 4,000 years ago.
The key discovery is a single one: do the same thing to both pans of a scale and the balance survives. With this one principle you can leave the unknown standing alone in any expression, however tangled.
Memorized rules like 'flip the sign when moving across' are all shadows of this scale surgery. See the principle, and the rules no longer need memorizing.
Insight
Insights from the video
โAn equation is a question; solving is scale surgery.โ
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.
โThe goal isn't computation โ it's isolation.โ
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.
Misconception
Common misconceptions
'The sign flips when a term moves across' is a rule to memorize.
It's not a rule โ it's a footprint. Subtract 3 from both sides and the +3 on the left vanishes while a โ3 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 โ an identity. 3x+2=11 is true only when x=3 โ a condition. That's an equation, and solving it means finding the moment it becomes true.
Formula
Writing it as math
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 โ like adding the same weight to both pans. (Division only by nonzero.)
What 'moving across' really is
The +2 didn't 'cross over' as โ2 โ 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 โ it's the definition of a solution.
In Real Life
Where you meet it in real life
Finding the pre-sale price
'$35 after a 30% discount' โ 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?' โ write the condition as an equation, isolate the unknown. Same structure.
Solving conditions in code
'When does the balance hit zero?' โ even in programming, the thinking is: set up the equation, then solve it.
Try Yourself
Test yourself
Q15x โ 7 = 18. Find x with exactly two scale moves.Show answer โพ
Add 7 to both sides โ 5x = 25. Divide both sides by 5 โ x = 5. No 'moving across' rule needed โ two scale moves finish it.
Q2Is 2x + 6 = 2(x + 3) an equation? Can you find x?Show answer โพ
Expand: 2x+6 = 2x+6 โ true for every x. That's an identity: the solution isn't 'one value' but 'all of them'. The reverse also exists โ 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' โ write the equation and solve it.Show answer โพ
2x โ 4 = x. Subtract x from both sides: x โ 4 = 0; add 4: x = 4. Translating words into an equation is half of solving.
Try answering yourself before revealing it โ getting it wrong is where learning starts.
Connection
Concepts connect
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.
โ 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 โ the view widens.
Go to the Functions lab โRelated
Labs worth exploring together
Related lab
Percent
Finding a pre-sale price IS solving an equation โ see 0.7x = 35 in the field.
Go to the Percent lab โ