Liri found an equation on her map: x + 5 = 12. “What does x have to be for that to be true?”
Erklärus: “Imagine a pair of scales. Both sides must always stay equal. Whatever you do on the left, you must do the same on the right.”
A balance made of numbers
⚖️ The balance level
An equation is like a balance that is level: the left side and the right side have the same value. The equals sign “=” is the pivot.
The balance principle
Whatever you do to one side of the equation, you must always do to the other side too – otherwise the balance tips over.
✓ Allowed
Both sides +3
Both sides ×5
Both sides −7
Both sides ÷2
✗ Forbidden
Only left +3
Only right ×5
Different values
÷ 0 (never!)
To get x on its own, you do the same on both sides:
x + 5 = 12
x + 5 − 5 = 12 − 5
x = 7
Erklärus: “We subtract 5 from both sides – the balance stays level, and x stands there on its own.”
Solve: x + 8 = 15 → x =
✖️ Multiplying and dividing too
Erklärus: “Solve: 3x = 15. Here x is multiplied by 3. What is the opposite action?”
Liri: “Dividing by 3 – on both sides!”
📐 Step by step
3x = 15Starting point
÷3 on both sides:3x ÷ 3 = 15 ÷ 3
Simplify:x = 5
Check:3 × 5 = 15 ✓
Solve: 4x = 28 → x =
🔗 Combined – first plus/minus, then times/divide
Erklärus: “Now together: 2x + 3 = 11. Which operation do you do first?”
Liri: “First get rid of the +3 – so −3 on both sides. Then 2x = 8 is left. Then ÷2. x = 4!”
📐 Two steps combined
2x + 3 = 11
Step 1, −3:2x + 3 − 3 = 11 − 3 → 2x = 8
Step 2, ÷2:2x ÷ 2 = 8 ÷ 2 → x = 4
Check:2×4 + 3 = 8 + 3 = 11 ✓
Solve: 3x + 2 = 14 → x =
🧮 Liri’s equation solver – interactive
Enter an equation of the form ax + b = c and see the steps of the solution:
⚖️ Equation of the form: ax + b = c
🌌 Equations in the universe
Scientists solve equations to find out how much fuel a rocket needs for a particular distance.
The famous formula E=mc² is also an equation – it describes how mass and energy are connected.
❌ Common mistake
Many people change only ONE side of the equation – then the balance is no longer right!
Tip: whatever you do on one side (add, subtract, multiply, divide), you must ALWAYS do on the other side too.
⚖️ Exercise 1
Solve: x + 6 = 20 → x = ?
➖ Exercise 2
Solve: x − 4 = 9 → x = ?
✖️ Exercise 3
Solve: 5x = 45 → x = ?
The balance stays level
Liri solved the equation from her map: x=7. At last she knew the missing value for her route. Then she remembered something quite real.
Liri: “A balance that is level… that is exactly how the engines were aligned at home. The same thrust always had to be on the left and on the right, otherwise the ship went off course.”
Erklärus: “A wonderful picture, Liri – and no coincidence. Engineers solve real equations in exactly this way, side by side in balance. Anyone who can do that can work out courses, align engines – and perhaps even their own way home.”
Liri: “As long as I keep the balance level, I can find every hidden number. And sometimes a piece of home too.”
“Solving equations: whatever you do on one side, you must do the same on the other. That way the equation stays true until x stands on its own.”
What you learned today
x + 5 = 12 → x = 7
Always treat both sides the same way
🌟
Chapter 43 mastered!
Liri can now solve simple equations – the balance always stays level.
Erklärus’s notebook – page 43:
“Solving equations: always treat both sides the same way (balancing method). Adding/subtracting/multiplying/dividing – always on both sides at the same time.”
“Solving equations: always treat both sides the same way (balancing method). Adding/subtracting/multiplying/dividing – always on both sides at the same time.”