Variables & Expressions

The name tag fell off

In the Bedrock stratum you built real numbers from scratch — $\mathbb{N}$, then $\mathbb{Z}$, then $\mathbb{Q}$, then $\mathbb{R}$. You know what those things ARE. Now suppose I hand you a number but tape a sticky note labelled $x$ over its face so you can't read it. That's a variable. The number is still there. It still obeys every law you derived. The only thing missing is the specific value.

This sounds trivially obvious, and it is — which is why it's the secret weapon of algebra. Because the laws are about STRUCTURE, not about specific values, every truth that holds for all numbers automatically holds for expressions built from variables. When the Laws of Arithmetic node told you that $a \cdot (b + c) = a \cdot b + a \cdot c$ for all real numbers $a$, $b$, $c$, it silently handed you the engine of all of algebra. Every manipulation you will ever do in this stratum is the distributive law, the commutative law, or the associative law from Bedrock, wearing a different costume.

Translating English into algebra

The Federation of Boring Textbook Authors wants you to translate "three less than twice a number" into $3 - 2x$ and just accept it. That's wrong, and you should be furious. Let's do it properly.

Let $n$ be the unknown number. "Twice a number" means $2 \cdot n$, i.e., $2n$. "Three less than" means you subtract three from that result: $2n - 3$. Done. The phrase describes a sequence of operations; translate each operation in the order described, left to right, and you cannot go wrong.

Common phrase-to-algebra dictionary:

• "the sum of $a$ and $b$": $a + b$
• "five more than $x$": $x + 5$
• "four times a number, decreased by seven": $4n - 7$
• "the quotient of $y$ and three": $y / 3$
• "the square of the sum of $x$ and two": $(x+2)^2$

The last one is crucial. Order and parentheses encode the STRUCTURE of the phrase. "$x^2 + 2$" is something completely different from "$(x+2)^2$". Get this wrong and every word problem you ever see will quietly hand you a wrong equation while you smile and nod.

Like terms and distributivity, naked

"Like terms" sounds like a grammar rule some dusty pedagogue invented. It isn't. It's the distributive law wearing jeans.

$3x + 5x = (3 + 5)x = 8x.$

That middle step IS distributivity — the law $a(b+c) = ab + ac$ run in reverse, with $x = a$, $3 = b$, $5 = c$. You're not "combining like terms" according to some mystical rule. You're factoring out $x$ because $x$ is a common factor, and then adding the coefficients because those are just numbers.

Consequence: you can combine $3x$ and $5x$ but you cannot combine $3x$ and $5x^2$. Why? Because $x$ and $x^2$ are different beasts; there's no single factor to pull out. Trying to add $3x + 5x^2$ is like trying to add 3 wolves and 5 Tuesdays — they share nothing.

This same distributive engine is why $(x+2)(x+3)$ expands the way it does — we'll dismantle that in the Polynomials lesson. File it away.

Evaluating expressions: parentheses are load-bearing

To evaluate $2x^2 - 3x + 1$ at $x = -4$: replace every $x$ with $(-4)$, parentheses included, ALWAYS.

$2(-4)^2 - 3(-4) + 1 = 2 \cdot 16 + 12 + 1 = 32 + 12 + 1 = 45.$

The parentheses around $-4$ are not decoration. Without them, $-4^2$ reads as $-(4^2) = -16$, which hands you the wrong answer and nobody warns you. Wrap every substituted value in parentheses, every time, as a physical reflex. This rule has saved more grades than any other single habit in the entire subject.

Expression vs equation: phrase vs sentence

An expression is a mathematical phrase: $2x + 3$, $x^2 - 5x + 6$, $\frac{3}{x-1}$. It doesn't claim anything; it just describes a quantity. You evaluate it, you simplify it, you stare at it.

An equation is a mathematical sentence: $2x + 3 = 11$. It makes a claim: "there exists a value of $x$ that makes the left side equal to the right side." You solve it. You prove it. The next lesson is entirely about equations.

The distinction matters because the tools are different. You simplify expressions; you solve equations. You never "solve" $2x + 3$ for anything — there's nothing to solve, it's a phrase. You never "simplify" $2x + 3 = 11$ — it already says exactly what it says.