Algebra
Equations, inequalities, expressions and functions. Many questions have a quick trick, and sometimes trying the choices is fastest.
Equations, inequalities, expressions and functions. Many questions have a quick trick, and sometimes trying the choices is fastest.
Equations
Linear equation
Isolate x: move numbers across with the opposite sign, then divide.
From words to an equation
“multiplied by” = ×, “added” = +, “subtracted” = −, “the result is” = =.
A number and its parts
Pick x = a common multiple of the bottoms so the fractions disappear.
Two equations: find x + y
Add or subtract the equations instead of finding each variable.
Quadratic equation
Factor into two brackets: find two numbers that multiply to the constant and add to the x-coefficient.
Absolute value
|x − a| = b gives two cases: x − a = b or x − a = −b.
Equations with fractions and roots
Root = number: square both sides. A sum of fractions: multiply by the common multiple of the bottoms. Fraction = fraction: cross-multiply.
Exponential equation
Write both sides with the same base, then equate the powers.
Proportion (cross-multiply)
a b = c d ⇒ a × d = b × c
A root in the equation
Square both sides to remove the root.
Expressions and identities
Substitution
Put the number in place of the variable in brackets, especially if it is negative.
Substituting two variables
Replace each variable with its value, then use the order of operations.
Difference of two squares
x² − y² = (x + y)(x − y)
Square of a sum / difference
(x ± y)² = x² ± 2xy + y²
Simplifying expressions
Expand the brackets (watch the minus sign in front) and combine like terms.
Power of a power
(a m) n = a m × n
Consecutive integers
The mean of consecutive numbers = the middle one = sum ÷ count.
Inequalities, functions and variation
Inequality
Solve like an equation, but flip the sign when multiplying or dividing by a negative.
Function
f(3) means put 3 in place of x.
Direct and inverse variation
Direct: y ÷ x is constant. Inverse: y × x is constant.
Try the choices: in most equations, plug the choices in starting with the middle one; if the result is too big, move to a smaller one.
Quick identities
From the sum and the product
x² + y² = (x + y)² − 2xy, 1 x + 1 y = x + y xy, x² + 1 x² = (x + 1 x)² − 2