The distributive law

Algebra, grade 7 Lesson 14 of 22Learning

The distributive law

To multiply a number by a sum, you can multiply it by each number in the sum and add the products: $\displaystyle {3\cdot(10+2)}={3\cdot10+3\cdot2}={30+6}={36}$ . This is the distributive law:

\displaystyle a(b+c)

\displaystyle =

\displaystyle ab+ac

Thanks to it, brackets can be opened even when they hold a variable: $\displaystyle {3(x+4)}={3x+12}$ .

The distributive law as an area

A rectangle is $\displaystyle x+4$ wide and 3 high.
Its area is $\displaystyle 3(x+4)$ . But the rectangle is made of two rectangles: the area of one is $\displaystyle 3x$ and of the other $\displaystyle 12$ . So $\displaystyle {3(x+4)}={3x+12}$ .

Distributing a difference and negative numbers

Careful: multiply every term in the brackets, not only the first.

\displaystyle 3(x+4)

is not

\displaystyle 3x+4

The distributive law holds for a difference and for negative numbers too. Multiply each term in the brackets, with its sign: $\displaystyle {5(2a-3)}={10a-15}$ ; $\displaystyle {-2(x+6)}={-2x-12}$ ; $\displaystyle {-3(2-y)}={-6+3y}$ .

A minus before brackets

A minus before brackets is multiplication by $\displaystyle -1$ : $\displaystyle {-(a-7)}={-a+7}$ . When the brackets are opened, the sign of every term inside flips: $\displaystyle {6-(2x-3)}={6-2x+3}={9-2x}$ .

Exercise 1

Open the brackets.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Distribute, then collect

First open the brackets, then collect like terms: $\displaystyle {4(x+2)+3x}={4x+8+3x}={7x+8}$ ; $\displaystyle {2(3x-1)-(x+5)}={6x-2-x-5}={5x-7}$ .

Exercise 2

Simplify: open the brackets, then collect like terms.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

At break, each of 4 friends bought a sandwich for $\displaystyle s$ ₪ and a bottle of water for 6 ₪.
Write two different expressions for the total they all paid, and show that they are equivalent.

Formulas: algebra, grade 7

Open as a page

Adding and subtracting signed numbers

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

Multiplying and dividing signed numbers

Positive times negative – negative
$\displaystyle a\cdot(-b)$ $\displaystyle =$ $\displaystyle -(a\cdot b)$
Negative times negative – positive
$\displaystyle (-a)\cdot(-b)$ $\displaystyle =$ $\displaystyle a\cdot b$
Dividing: different signs – the quotient is negative
$\displaystyle (-a):b$ $\displaystyle =$ $\displaystyle -(a:b)$ $\displaystyle b\neq0$
Dividing: the same sign – the quotient is positive
$\displaystyle (-a):(-b)$ $\displaystyle =$ $\displaystyle a:b$ $\displaystyle b\neq0$

Powers and roots

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

The laws of arithmetic and like terms

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

The distributive law

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

Distance, speed and time

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