Substitution
A taxi fare rule, a recipe scaled up for a bigger crowd, a temperature converted from °C to °F —
a formula only becomes a real, usable number once you know the values to drop into it. That
plugging-in is substitution, and it is how every formula finally earns its keep.
Once you can turn
words into an expression,
a letter stands in for a number you don't yet know. Substitution is the
step where you finally do know it: you replace each letter with its given value and
work the expression out like ordinary arithmetic.
Say x = 4. Then everywhere you see an x,
write 4 instead:
3x + 2 = 3 \times 4 + 2 = 12 + 2 = 14
Notice the last two steps follow the
order of operations
— the multiplication 3 \times 4 happens before the
+ 2. Substitution doesn't change those rules; it just gives you
numbers to apply them to.
Multiply before you add
The most common slip is forgetting that a letter next to a number means
multiply. 2n is short for 2 \times n.
So when n = 4:
2n = 2 \times 4 = 8
It is not 24 — you don't just glue the
2 and the 4 together. And in a bigger
expression you still do the multiplying first:
2n + 1 = 2 \times 4 + 1 = 8 + 1 = 9
Two traps that catch everyone:
- 2n with n = 4 is
2 \times 4 = 8 — not 24.
The letter sits next to the number, so it means multiply, not write-side-by-side.
- Do the multiplication before the addition. In
2n + 1 you work out 2 \times 4 = 8
first, then add 1 to get 9 — never
4 + 1 = 5 doubled.
A stall sells apples at 3p each and charges
2p for the bag. The cost of n apples is
a little machine: \text{cost} = 3n + 2. Put in a number of apples,
and out comes a price. Buy 4 apples and you substitute
n = 4: 3 \times 4 + 2 = 14p. Buy
10 and it is 3 \times 10 + 2 = 32p.
One formula, any number of apples.
One letter at a time
Press play to watch a letter get replaced by a number, then the expression collapse to a
single answer (a fresh value each time):
Substituting into a formula
A formula is just an expression with a name. Evaluating it is exactly the
same job: plug the measurements in for the letters and compute. The perimeter of a rectangle
(the distance all the way round) is
P = 2l + 2w
where l is the length and w is the
width. For a rectangle 5 cm long and 3 cm
wide, substitute l = 5 and w = 3:
P = 2 \times 5 + 2 \times 3 = 10 + 6 = 16 \text{ cm}
Substitute every letter, then let the order of operations finish the job — both
multiplications first, then the addition.
On a farm with c cows and d ducks, how
many legs are there? Each cow has 4 legs and each duck has
2, so the formula is
\text{legs} = 4c + 2d. With 3 cows and
5 ducks, substitute c = 3 and
d = 5: 4 \times 3 + 2 \times 5 = 12 + 10 = 22
legs. The formula does the counting for any farm you like.
See it: the substitution machine
Think of an expression as a machine. You feed a number in at the top for
n; the machine multiplies and adds by its rule; a single answer
drops out at the bottom. Press Refresh for a new number, then step through to
watch it multiply first, then add.
More than one letter
With several letters, substitute them all, then evaluate. If a = 5
and b = 2:
a^2 - b = 5^2 - 2 = 25 - 2 = 23
Be careful with the powers and any
negative numbers
— substitute first, then let the order of operations finish the job. This is exactly how you
evaluate a formula: plug the measurements in for the letters and compute.
Khan Academy walks through evaluating an expression by substitution here: