2x+3 · Variables & Expressions
A placeholder for the unknown — the variable.
A pouch holds an unknown number of candies. Add 3 more — how many should you say there are?
Read the role of a variable.
Arithmetic that works even when you don't know the number.
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Why does this question matter?
For ages mathematics handled only known numbers. But real problems mostly begin with unknowns — the candies in the pouch, next year's age, the price we seek.
The breakthrough: give the unknown a name (x) and a seat. Write 'pouch+3' as x+3 and you can build, tidy, and compare without ever knowing the value.
With this invention mathematics became a language. Formulas (area = length×width), equations, functions — everything after is built on letters.
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 — you don't know how many candies the pouch holds. Add 3 — how many are there?
Build an expression
Call the pouch's candy count x. Use the buttons to build.
x
expression so far
Now open the pouch — suppose it held 5.
Read the idea from another angle
x is a seat, not an answer.
See x only as 'the answer to find' and algebra feels scary. x is just a chair for a number not yet known — set out the chair and the computation works for whoever sits down later.
The expression itself can be the answer.
Arithmetic always ended in a single number; in algebra, 3x+2 is a finished answer. Crossing this gate is the real moment one passes from arithmetic to algebra.
Now read the definitions with their conditions
What the pouch experiment confirmed, in mathematical language.
Using letters
Name the unknown and computation starts without it. 2x means 2×x — the multiplication sign is omitted.
Substitution
The moment the pouch opens — put a number in the letter's seat and the value computes.
Collecting like terms
Pouches with pouches, loose candies with loose candies — only like terms combine.
Make one more distinction
2x means 'a 2 written next to an x', so if x=5 then 2x=25.
2x is 2×x with the multiplication sign omitted — x=5 gives 2×5=10. The number in front (the coefficient) is always glued on by multiplication.
3x + 2 should compute further, to 5x or 5.
Three pouches and two loose candies can't merge — while x is unknown, 3x+2 IS the final answer. Only like terms combine. Getting comfortable with 'the expression itself is the answer' is algebra's first step.
Where else can this idea help?
Every formula is an expression
Area = length × width (ab), speed = distance ÷ time (d/t) — a formula is an expression that works for any numbers. Letters let us write once and reuse forever.
Fares and bills
Taxi fare 0.8x+4.8 (x in km) — behind every bill and rate table sits an expression. The pattern lab's □, grown into x.
Spreadsheets and code
Excel's =A1*2+3, code's price * n — cell names and variables ARE letters. The spreadsheet is humanity's most-used algebra tool.
Age puzzles
'In 10 years, dad will be twice the son's age' — set the son's age to x and the sentence becomes an expression, then an equation.
Explain it in your own words
Try another question
Q1Pencils cost $1.20 each. Write the cost of x pencils plus one $0.50 eraser.Show answer ▾
1.2x + 0.5 (dollars). Substitute x=3 and get $4.10 — one expression is a price tag for every quantity.
Q2Simplify 5x − 2x + 7 − 3.Show answer ▾
3x + 4. Pouches with pouches (5−2=3), loose with loose (7−3=4) — collecting like terms.
Q3If x = −2, what is 3x + 10?Show answer ▾
3×(−2)+10 = −6+10 = 4. Wrap negatives in parentheses when substituting and mistakes vanish.
Try answering yourself before revealing it — getting it wrong is where learning starts.
What question comes next?
A reading path containing this lesson: Unknown numbers and balance · 1/3
Previous concept
Patterns & Correspondence
Writing relations with □ and △ is the soil variables grow from.
← Patterns & Correspondence labLeads to next
Equations
Now that you can write expressions, ask the question — 'for which x are both sides equal?' The equation.
Go to the Equations lab →Related lab
Patterns & Correspondence
□×4=△'s □ grew up to become x — the variable's ancestor.
Go to the Patterns & Correspondence lab →Related lab
Equations
Start asking when two expressions become equal, and you have an equation.
Go to the Equations lab →