Foundations & Expressions

A1.A.1Properties of real numbers01/05

Properties of real numbers.

Step01of 05

The properties below aren't trivia — they're the rules that justify EVERY algebraic move you'll make this year. They feel obvious because they ARE obvious; the point is naming them so you can cite the right rule.

Commutative — order doesn't matter for $+$ and $\cdot$ .

x + y = y + x x \cdot y = y \cdot x

Subtraction and division are NOT commutative: $5 - 2 \neq = 2 - 5$ .

A1.A.1 — Properties of real numbers.

Step 01 of 05

Commutative — order doesn't matter for $+$ and $\cdot$ .

x + y = y + x x \cdot y = y \cdot x

Subtraction and division are NOT commutative: $5 - 2 \neq = 2 - 5$ .

Step 02 of 05

Associative — grouping doesn't matter for $+$ and $\cdot$ .

(x + y) + z = x + (y + z)

(x \cdot y) \cdot z = x \cdot (y \cdot z)

This is what lets you compute $2 + 3 + 7$ as $5 + 7$ or $2 + 10$ — your choice.

Step 03 of 05

Distributive — multiplication distributes over addition.

a (x + y) = a x + a y

The most-used property in Algebra I. It powers expanding (going right) AND factoring (going left). Example:

3 (x + 4) = 3 x + 12

Step 04 of 05

Identity — adding 0 or multiplying by 1 leaves the value unchanged.

x + 0 = x x \cdot 1 = x

Inverse — every number has a partner that undoes it.

x + (- x) = 0 x \cdot \frac{1}{x} = 1 (x \neq = 0)

Step 05 of 05

Why this matters in practice.

You rearrange a sum	→commutative + associative
You expand or factor	→distributive
You "add zero" to complete a square	→additive identity + inverse
You divide both sides	→multiplicative inverse

Key insight

Every legal algebra step traces back to one of these properties. When a step feels wrong, name the property you'd have to invoke — if you can't, the step probably IS wrong.

A1.A.2 — Combine like terms and distribute.

The expression we'll simplify

3 (2 x + 5) - 4 x + 7

Step 01 of 05

Like terms have the SAME variable raised to the SAME power. The coefficient (the number out front) can differ.

$3 x$ and $- 4 x$	→like terms
$5 x^{2}$ and $2 x$	→NOT like (different power)
$5$ and $7$	→like (both constants)
$3 x y$ and $- 2 x y$	→like (same variable mix)

Step 02 of 05

Step 1 — distribute first. Multiply the $3$ across the parentheses. The minus sign in front of $4 x$ stays put — it's not inside parens.

3 (2 x + 5) - 4 x + 7

= 6 x + 15 - 4 x + 7

Step 03 of 05

Step 2 — group like terms. Slide the x-terms next to each other and the constants next to each other (commutative property earns its keep here).

= (6 x - 4 x) + (15 + 7)

Step 04 of 05

Step 3 — combine each group.

= 2 x + 22

That's it. The expression went from 4 terms to 2 — fully simplified.

Step 05 of 05

Watch for the negative-distribution trap. A leading minus is really a $- 1$ ; it flips every sign inside.

- (2 x - 5) = - 2 x + 5 (not - 2 x - 5)

Key insight

Order: distribute → combine. If you try to combine before distributing, you'll bundle terms that look alike but really aren't — those parentheses are still in the way.

A1.A.3 — Evaluate expressions with multiple variables.

Evaluate this expression

2 x^{2} + 3 x y - y at x = 3, y = - 2

Step 01 of 05

Step 1 — substitute, with parentheses. Wrap each value in parens the moment it goes in. This is the single best habit for avoiding sign errors with negatives.

2 (3)^{2} + 3 (3) (- 2) - (- 2)

Step 02 of 05

Step 2 — exponents next. Order of operations: parens, exponents, then multiply/divide, then add/subtract (PEMDAS).

2 (9) + 3 (3) (- 2) - (- 2)

$(3)^{2} = 9$ . The $2$ out front waits — the exponent binds only to the $(3)$ .

Step 03 of 05

Step 3 — multiplications, left to right.

= 18 + (- 18) - (- 2)

$2 \cdot 9 = 18$ . $3 \cdot 3 \cdot (- 2) = - 18$ . The last term $- (- 2)$ stays parked for now — it's an add, not a multiply.

Step 04 of 05

Step 4 — clean up the signs. A minus in front of a negative flips it: $- (- 2) = + 2$ .

= 18 - 18 + 2

Step 05 of 05

Step 5 — add left to right.

= 0 + 2 = 2

Key insight

Two habits make multi-variable evaluation safe: (1) wrap every substituted value in parens immediately, (2) follow PEMDAS strictly — exponents BEFORE the coefficient out front. Those two habits eliminate ~90% of sign errors.

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