The coordinate system in the first quadrant

To say exactly where a point is on the page, we use two perpendicular number lines: the $\displaystyle x$ -axis (across) and the $\displaystyle y$ -axis (up). They meet at the point $\displaystyle (0,0)$ , the origin.

An ordered pair

First

\displaystyle x

, then

\displaystyle y

– as in the alphabet.

Every point has two coordinates: its $\displaystyle x$ -coordinate (how many steps right of the origin) and its $\displaystyle y$ -coordinate (how many steps up). They are written in brackets, $\displaystyle x$ first, as an ordered pair: the point $\displaystyle A$ is $\displaystyle (3,2)$ .

The order matters: $\displaystyle {B}={(2,3)}$ is a different point from $\displaystyle {A}={(3,2)}$ . A point on the $\displaystyle x$ -axis has height 0, like $\displaystyle {C}={(6,0)}$ , and a point on the $\displaystyle y$ -axis has $\displaystyle x$ -coordinate 0, like $\displaystyle {D}={(0,4)}$ .

Exercise 1

Write the coordinates of each point.

Exercise 2

Mark the points $\displaystyle {L}={(2,5)}$ , $\displaystyle {M}={(5,2)}$ , $\displaystyle {N}={(0,1)}$ and $\displaystyle {O}={(4,0)}$ in a coordinate system.
Start by drawing the axes on the squares.

The length of a segment

$\displaystyle {P}={(1,4)}$ and $\displaystyle {Q}={(6,4)}$ have the same $\displaystyle y$ -coordinate, so the segment $\displaystyle PQ$ is horizontal. Its length is the difference of the $\displaystyle x$ -coordinates: $\displaystyle {6-1}={5}$ .

$\displaystyle {Q}={(6,4)}$ and $\displaystyle {R}={(6,1)}$ have the same $\displaystyle x$ -coordinate, so the segment $\displaystyle QR$ is vertical, and its length is $\displaystyle {4-1}={3}$ .

Exercise 3

$\displaystyle A$ , $\displaystyle B$ and $\displaystyle C$ are three vertices of the rectangle $\displaystyle ABCD$ .
What are the coordinates of the fourth vertex, $\displaystyle D$ ?
How long are the sides of the rectangle?
Calculate its area.

\displaystyle AB

/ – 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

\displaystyle BC

/ – 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

Area

/ – 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 4

$\displaystyle (2,1)$ and $\displaystyle (5,1)$ are two neighbouring vertices of a square.
Draw such a square.
What are the coordinates of the other two vertices?
There are two possibilities – find both.

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
$\displaystyle s$ $\displaystyle =$ $\displaystyle v\cdot t$