Substitution

A formula is a machine waiting to start, and letters are its empty input slots. Variables and the Function Machine showed that $f(x) = 3x + 1$ is written once yet handles every input. Feeding in a concrete number and computing the result is called substitution.

Substitution: swap letters for numbers

Take the expression $a^2 + 2b$ . When $a = 3$ and $b = 4$ , what is it worth? Three steps:

Copy the expression: $a^2 + 2b$ ;
Swap in the values: $3^2 + 2 \times 4$ ;
Compute in order: $9 + 8 = 17$ .

Copy first, swap second, never skipping a step — that is the steadiest rhythm for substitution. Mental short-cuts drop terms; forgetting the $b$ inside $2b$ is the classic slip.

Order of operations still applies

Letters change nothing about the order of operations: powers first, then multiplication and division, then addition and subtraction.

Take $2x^2$ with $x = 3$ :

Correct: $2 \times 3^2 = 2 \times 9 = 18$ ;
Wrong: first $2 \times 3 = 6$ , then square to get 36.

$2x^2$ means $2 \times (x^2)$ — the square grabs only the $x$ , not the 2 in front.

Two high-frequency errors

$2a$ means $2 \times a$ . If $a = 5$ , then $2a = 10$ — not the two-digit number "2a", and not $2 + a = 7$ .
Squaring a negative number needs brackets: $(-3)^2 = (-3) \times (-3) = 9$ , but without brackets $-3^2$ means $-(3^2) = -9$ . One missing pair of brackets flips the sign.

Formulas from real life

Formulas are everywhere, and substitution is how you use them. The circumference of a circle (the star of Circles) is

$C = 2\pi r$

For a circle of radius 3 cm, taking $\pi \approx 3.14$ :

$C = 2 \times 3.14 \times 3 = 18.84$

so the circumference is about 18.8 cm. What if the radius doubles? At $r = 6$ you get $C = 37.68$ — doubled too. Substitute a few more values and the formula starts feeling like an old friend.

The Substitution Machine

InteractiveSubstitution Machine

A secret rule hides inside the machine. Feed it numbers and guess the rule!

? ? ?

Output

—

In Variables and the Function Machine you used this machine to guess the rule backwards. Flip the game around: first evaluate the rule in your head, then feed the machine and compare. When your mental substitution matches the machine every time, the skill has truly landed.

Check yourself

Quick quiz

1. With a=3 and b=4, what is a²+2b?
2. With x=4, what is 2x²?
3. What is (-3)²?