Liri found a formula in an old star atlas: “x + 5 = distance to the nearest star”. “What is x?”
Erklärus: “x is a variable – a placeholder for a number that we don’t know yet or that can change.”
A number that hides
🔤 What is a variable?
Imagine you don’t know a number yet – perhaps you will only know it later, or it can even change. So that you can already calculate with it, you give it a placeholder: a letter, usually x.
Erklärus: “A variable is simply a letter that stands for a number. If you later put in a definite number for x, that is called substitution.”
📦 Variable
x, y, a, b, n…
letter = unknown number
🔢 Coefficient
3x → 3 is the coeff.
The number in front of the variable
🪨 Constant
3x + 5 → 5 is the const.
Fixed value, never changes
🔍 Substituting
x=4: 3×4+5 = 17
Replace the variable with a number
Liri: “And what if there is no number in front of the x at all, as in ‘x + 5’?”
Erklärus: “Then the coefficient is secretly a 1. You simply write 1×x as x – you leave out the 1 because it changes nothing.”
Expression: 2x + 3. Substitute x=5.
✂️ Adding, subtracting and simplifying expressions
An expression is a calculation made of numbers, variables and operation signs. You can collect like terms (the same variable) – just as you add up like things.
Erklärus: “Liri, what is 3x + 5x? Remember what x means.”
Liri: “If x is an unknown number… then 3x is three of that number, and 5x is five of them. Together that is 8 of that number – so 8x!”
🍎 Line of thought – like terms
3 apples + 5 apples = 8 apples – “like” means: the same variable.
3x + 5x = (3+5)x = 8x
Simplify: 4x + 3x = ?x
🍐 Unlike terms – apples stay apples
Erklärus: “And what is 3x + 5y? Are those like terms?”
Liri: “x and y are different unknown numbers… so 3x is a kind of apple and 5y a kind of pear. You can’t combine apples and pears – it stays 3x + 5y.”
⚠️ Common mistake
Wrong: 3x + 5y = 8xy
Right: 3x + 5y stays 3x + 5y – it can’t be simplified!
🔓 Expanding brackets – the distributive law
Erklärus: “What is 3 × (x + 4)? The factor on the outside is multiplied by each term in the brackets separately.”
Liri: “I multiply 3 by x… that is 3x. And 3 times 4… that is 12. So 3x + 12!”
🧠 The distributive law – made visible
3 × (x + 4) = 3x + 12
a × (b + c) = a×b + a×c
Expand the brackets: 2 × (x + 3) = 2x + ?
🔍 Substituting variables – from the general to the concrete
To evaluate an expression means: substitute a definite number for the variable and then calculate as normal.
📍 Example: 4x + 3 for x = 5
4x + 3 | substitute x = 5
= 4 × 5 + 3
= 20 + 3
= 23
🔣 Evaluating variables – interactive
Work out the expression ax + b for different values:
—
Liri: “That is like the travel formula from before! If I substitute d=300 and v=50 into t=d÷v, I get t=6. Six hours of travel time!”
Erklärus: “Exactly. And if the distance changes, the formula stays the same. Only the number you substitute changes. That is the power of variables.”
🌌 Variables in the universe
Physicists use variables such as t (time), v (speed) or m (mass) to write down universal formulas that apply to every rocket.
A formula such as “distance = v × t” works for any speed and time at all – that is the power of variables.
❌ Common mistake
Different variables must NOT be combined: 3x + 2y stays 3x + 2y, it does NOT become 5xy!
Tip: you may only add or subtract the same variables (the same letter).
🔤 Exercise 1 – Substituting
Expression: 5x - 3. Substitute x=4.
✂️ Exercise 2 – Simplifying
Simplify: 7x - 2x = ?x
🚀 Exercise 3 – Universe
Distance = v × t. With v=6 and t=8 – how far does the rocket fly?
A formula for every distance
Liri wrote a formula of her own for her journey: distance = v × t. No matter how fast she flew, the formula would always be right. Then something occurred to her.
Liri: “Wait – that is how we worked out the landing position at home, too! Not with a fixed number, but with a formula that remembered letters instead of numbers. The same formula for every ship, only different values put in.”
Erklärus: “That is not a memory from just anywhere, Liri – that is real knowledge from your home. Anyone who understands variables can use a single formula for a thousand different situations. Perhaps this very formula will lead you back one day.”
Liri: “I don’t have to work out every journey separately any more. One formula is enough for all of them – and a piece of my memory has just come back.”
“Variable: placeholder for a number. Expression: a calculation with variables. Collect like terms (the same variable): 3x+5x=8x. Expand brackets: a×(b+c)=a×b+a×c.”
What you learned today
3x + 2x = 5x
Only combine the same variables
🌟
Chapter 42 mastered!
Liri can now calculate with variables – formulas that work for any number.
Erklärus’s notebook – page 42:
“Variables: letters as placeholders for numbers. Evaluating expressions: substitute values. Simplifying: only combine the same variables (3x+2x=5x).”
“Variables: letters as placeholders for numbers. Evaluating expressions: substitute values. Simplifying: only combine the same variables (3x+2x=5x).”