The cake scale conspiracy
Pip removes the weights from just one side. The scales file a complaint.
The formula in this story
Story 1 of 8
What you will take away
Explain why subtracting the same number keeps an equation balanced.
Read one piece at a time. There is no timer. “Help” stays beside you throughout.
Before reading, how does it feel? (optional)
See the five reading steps
Choose a section directly
What do the words mean?
An equation states that two expressions have the same value. A solution is a value of the unknown that makes the equation true.
In x + b = s, b and s are given numbers and x is the unknown. The experiment lets you choose a trial x and b; it sets the right side to their sum.
What are real numbers?
Real numbers are the numbers on the number line: integers, fractions and numbers such as √2. Our formulas apply to these numbers, even when the cake picture uses only positive lengths.
No other case needs to be read first.
A suspicious plan
Pip: “One mystery cake and three little cakes balance eight little cakes. I shall eat the three. On this side only.”
Remove the same signed amount from both sides. Pip is not allowed to snack on just one side.
The clue
Mika catches the plate. “Both sides lose three, detective. Fairness is the whole mechanism.”
The extra terms cancel. The mystery box is now alone: that is the value of x.
The twist
Five cakes remain opposite the mystery cake. Pip labels the evidence “breakfast.” Mika relabels it “x = 5.”
Put the answer back in the original equation. Both sides must still agree.
Change the evidence
Try examples to build intuition. Checking numbers is not the proof for every value.
Step 1 of 3
Subtract b from both equal quantities: (x + b) − b = s − b.
The diagram illustrates the displayed argument. Numerical examples do not replace its hypotheses or prove every case.
The proof, with all its steps
Claim and conditions
For real x, b and s, x + b = s holds exactly when x = s − b. In the story, b = 3 and s = 8.
- Subtract b from both equal quantities: (x + b) − b = s − b.
- The left side simplifies to x, so x = s − b. For the original story, 8 − 3 = 5.
- Check the reverse: substitute s − b for x. Then (s − b) + b = s. Both directions hold; the story example gives 5 + 3 = 8.
Pip has one more idea…
Pip wants to multiply both sides by zero to discover x. Does that determine it?
Read the explanation for Yes: whatever keeps equality also keeps all information.
Multiplication by zero preserves equality but loses information. Every x now satisfies 0 = 0; we cannot reverse that step.
Read the explanation for No: 0 = 0 has forgotten which x we started with.
Exactly. Adding or subtracting the same number is reversible. Multiplication by zero is not.
What makes more sense now?
Two small checks, then your own reflection. You can use the hints and try again. This records these questions, not a score for all of mathematics.
How does it feel now? (optional)
Your record for this story
Your first answers and your latest answers will appear here.
Where does your curiosity go next?
No score, no locked doors. You can return to any definition.
These are educational arguments and numeric experiments, not live Lean verification. The linked knowledge pages distinguish formal source excerpts from explanations.