Mathematics from zero
Expressions
“Three apples and two more” is a calculation: 3 + 2. “Some apples and two more” is
also a calculation — n + 2 — you just cannot finish it until you know n. That
unfinished-but-meaningful piece of maths is an expression.
After this lesson you can say what an expression is, see how it differs from an equation, evaluate an expression by substituting a value, identify its terms, and simplify it by combining like terms.
An expression combines numbers, variables, and operations. n + 2, 3 × x,
4x + 1 are all expressions. An expression has a value, but it carries no
equals sign — it does not claim two things are equal. It is a recipe for a number, not
a statement about one.
Evaluate an expression by substituting a value for the variable. An expression
with a variable does not have a single value until you choose one. To evaluate
4x + 1, pick a value for x — say x = 3 — put 3 everywhere x appears, then
compute: 4 × 3 + 1 = 13. A different x gives a different value.
An expression is built from terms. A term is a piece of the expression
separated from the others by + or −. In 4x + 1, the terms are 4x and 1. In
2n + 3n + 5, the terms are 2n, 3n, and 5. The number stuck to a variable —
the 4 in 4x — says how many of that variable the term holds.
Combine like terms to simplify. Terms that hold the same variable are like
terms, and they can be merged. 2n and 3n are like terms: two ns plus three
ns is five ns, so 2n + 3n = 5n. But 2n and 5 are not like terms — one counts
ns, the other is a plain number — so they cannot be merged.
Simplify 2x + 3 + 4x, then evaluate the result at x = 5.
First, simplify by combining like terms. The like terms are 2x and 4x — both
count xs. Two xs plus four xs is six xs: 2x + 4x = 6x. The 3 is a plain
number with no like term, so it stays. The simplified expression is 6x + 3.
Now evaluate at x = 5: substitute 5 for x. 6 × 5 + 3 = 30 + 3 = 33.
So the expression is 6x + 3, and at x = 5 its value is 33.
Why this works
Why can you only combine like terms? Because 2x means “two of whatever x is” and
5 means “five ones”. Adding them would be like adding two bags of an unknown weight
to five single coins and calling it “7” of something — seven of what? Only terms
measuring the same thing can be counted together. Like terms share a unit; unlike
terms do not.
Common mistake
A common mistake is reading 2n as 2 + n. It is not — 2n means 2 × n, two times
the variable. A number written directly against a variable always means multiply. So
at n = 4, 2n is 8, not 6. When in doubt, put the multiplication sign back in.
Evaluate the expression n + 7 at n = 5. Type the value.
Evaluate the expression 3n at n = 4. Type the value.
Evaluate the expression 2n + 1 at n = 6. Type the value.
Combine the like terms 2n + 3n. Type the number that ends up in front of n.
Evaluate the expression 5x − 2 at x = 4. Type the value.
Why can 2x + 3x be combined into 5x, but 2x + 3 cannot be combined?
An expression combines numbers, variables, and operations into a recipe for a value, and unlike an equation it carries no equals sign. Evaluate an expression by substituting a number for the variable and computing. An expression is built from terms, separated by + and −. Like terms — terms holding the same variable — can be combined to simplify; unlike terms cannot, because they measure different things.