Algebra rules cheat sheet

Every legal move in algebra comes from one of the laws below. When a manipulation feels risky, check it against this list, if it isn't here in some form, it probably isn't allowed.

Practise algebra

Basic laws

$a + b = b + a, ab = ba$ Commutative
$(a + b) + c = a + (b + c)$ Associative
$a (b + c) = ab + a c$ Distributive
$a + 0 = a, a \cdot 1 = a$ Identities
$a + (- a) = 0, a \cdot \frac{1}{a} = 1 (a \neq = 0)$ Inverses

Fractions

$\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}$
$\frac{a}{b} \cdot \frac{c}{d} = \frac{a c}{b d}$
$\frac{a / b}{c / d} = \frac{a}{b} \cdot \frac{d}{c}$
$\frac{a c}{b c} = \frac{a}{b} (c \neq = 0)$ Cancel common factors only

Special products

$(a \pm b)^{2} = a^{2} \pm 2 ab + b^{2}$
$a^{2} - b^{2} = (a - b) (a + b)$
$a^{3} \pm b^{3} = (a \pm b) (a^{2} \mp ab + b^{2})$
$(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$

Absolute value & inequalities

$∣ x ∣ = a ⟺ x = \pm a (a \geq 0)$
$∣ x ∣ < a ⟺ - a < x < a$
$∣ x ∣ > a ⟺ x < - a or x > a$
$a < b \Rightarrow - a > - b$ Multiplying by a negative flips the sign

Lines

$m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ Slope
$y = m x + b$ Slope-intercept
$y - y_{1} = m (x - x_{1})$ Point-slope
$m_{1} m_{2} = - 1$ Perpendicular lines

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