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Formulas: algebra, grade 7

Adding and subtracting signed numbers

  • The sum of two opposite numbers
    (−a)+a\displaystyle (-a)+a =\displaystyle = 0\displaystyle 0
  • Subtracting a number is adding its opposite
    a−b\displaystyle a-b =\displaystyle = a+(−b)\displaystyle a+(-b)
  • Subtracting a negative number
    a−(−b)\displaystyle a-(-b) =\displaystyle = a+b\displaystyle a+b

Multiplying and dividing signed numbers

  • Positive times negative – negative
    a⋅(−b)\displaystyle a\cdot(-b) =\displaystyle = −(a⋅b)\displaystyle -(a\cdot b)
  • Negative times negative – positive
    (−a)⋅(−b)\displaystyle (-a)\cdot(-b) =\displaystyle = a⋅b\displaystyle a\cdot b
  • Dividing: different signs – the quotient is negative
    (−a):b\displaystyle (-a):b =\displaystyle = −(a:b)\displaystyle -(a:b)b≠0\displaystyle b\neq0
  • Dividing: the same sign – the quotient is positive
    (−a):(−b)\displaystyle (-a):(-b) =\displaystyle = a:b\displaystyle a:bb≠0\displaystyle b\neq0

Powers and roots

  • A power – a\displaystyle a the base, n\displaystyle n the exponent: how many times a\displaystyle a is multiplied
    an\displaystyle a^{n} =\displaystyle = a⋅a⋅…⋅a⏟n\displaystyle \underbrace{a\cdot a\cdot\ldots\cdot a}_{n}
  • A number to the power 1 is the number itself
    a1\displaystyle a^{1} =\displaystyle = a\displaystyle a
  • A negative base with an even exponent – the power is positive
    (−a)2\displaystyle (-a)^{2} =\displaystyle = a2\displaystyle a^{2}
  • The square root – the positive number (or 0) whose square is the number under the root
    a2\displaystyle \sqrt{a^{2}} =\displaystyle = a\displaystyle aa≥0\displaystyle a\ge0

The laws of arithmetic and like terms

  • The commutative law of addition
    a+b\displaystyle a+b =\displaystyle = b+a\displaystyle b+a
  • The commutative law of multiplication
    a⋅b\displaystyle a\cdot b =\displaystyle = b⋅a\displaystyle b\cdot a
  • The associative law of addition
    (a+b)+c\displaystyle (a+b)+c =\displaystyle = a+(b+c)\displaystyle a+(b+c)
  • The associative law of multiplication
    (a⋅b)⋅c\displaystyle (a\cdot b)\cdot c =\displaystyle = a⋅(b⋅c)\displaystyle a\cdot(b\cdot c)
  • A number times itself – squared
    x⋅x\displaystyle x\cdot x =\displaystyle = x2\displaystyle x^{2}
  • Collecting like terms – add the coefficients; the letters stay
    ax+bx\displaystyle ax+bx =\displaystyle = (a+b)x\displaystyle (a+b)x

The distributive law

  • The distributive law
    a(b+c)\displaystyle a(b+c) =\displaystyle = ab+ac\displaystyle ab+ac
  • The distributive law over a difference
    a(b−c)\displaystyle a(b-c) =\displaystyle = ab−ac\displaystyle ab-ac
  • A minus before brackets – every term inside changes its sign
    −(a−b)\displaystyle -(a-b) =\displaystyle = −a+b\displaystyle -a+b

Distance, speed and time

  • s\displaystyle s the distance, v\displaystyle v the speed, t\displaystyle t the time
    s\displaystyle s =\displaystyle = v⋅t\displaystyle v\cdot t