Maths- Sequences, Subsitution, Rearranging and Linear Equations Flashcards
A number pattern which increases (or decreases) by the same amount each time is called a linear sequence. The amount it increases or decreases by is known as the common difference.
To find the common difference, you need to know how much the terms are increasing or decreasing by from one term to the next.
Sometimes, rather than finding the next number in a linear sequence, you want to find the 41st number, or 110th number, say.
Writing out 41 or 110 numbers takes a long time, so you can use a general rule.
So the sequence of numbers in the 5 times table has a common difference of 5 and an nth term of 5n.
But what happens if things get more complicated?
The common difference is still 5, but it’s not the 5 times table.
The 5 times table is 5, 10, 15, …
The sequence is 7, 12, 17, …
Each term in the sequence is 2 more than the corresponding term in the 5 times table, so the nth term is 5n + 2.
Highest Common Factor (HCF)
The factors of 12 are 1, 2, 3, 4, 6 and 12
The factors of 18 are 1, 2, 3, 6, 9 and 18
1, 2, 3 and 6 are factors of both 12 and 18.
They are known as the common factors of 12 and 18.
So the Highest Common Factor (HCF) of 12 and 18 is 6.
Lowest Common Multiple (LCM)
The multiples of a number are all the numbers that it will divide into.
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, …
The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, …
12, 24, and 36 are multiples of both 4 and 6 and are known as the common multiples of 4 and 6.
The lowest number that is a multiple of 4 and 6 is 12. So the LCM of 4 and 6 is 12.
Substitution
When letters in a formula are replaced by numbers, it is called substitution.
Example - Time
For the purpose of measuring time, the Earth’s surface is divided into 24 equal wedges of 15°, each called time zones and beginning at Greenwich, London (GMT). As you pass over each zone to the east you add 1 hour to GMT, and as you pass over each zone to the west you subtract 1 hour from GMT.
On this basis, call the time in London ‘g’.
The formula for working out the time in Bangkok, Thailand is g + 7
And the formula for working out the time in Santiago, Chile, is g - 4
These formulae allow us to substitute ‘g’ for any time in London to find out the time in Bangkok or Santiago.
Example - Temperature
Here is the formula to convert the temperature in degrees Fahrenheit (°F) to the temperature in degrees Celsius (°C)
c = 5(f - 32) / 9
where f represents the temperature in °F.
This formula allows you to substitute any °F temperature in for f to find its equivalent temperature in °C
Example
To find the temperature in °C when it is 68°F, substitute 68 for the f in the formula.
When f = 68,
5(f - 32) ÷ 9
= 5(68 - 32) ÷ 9
Remember to work out any calculation in brackets first: (68 - 32) = 36.
A number next to anything in brackets means the contents of the brackets should be multiplied, so 5(36) means 5 × 36:
5(36) ÷ 9
= (5 × 36) ÷ 9
= 180 ÷ 9 = 20
So 68°F = 20°C
Changing the subject of a formula
Sometimes we will need to rearrange a formula to find the value of a subject.
We may know the area of a circle and need to find the radius. To do this, we rearrange the formula to make the radius the subject.
The area of a circle (A) is πr2. So:
A = πr2
We will now rearrange the formula to make ‘r’ the subject.
A = πr2
Start by dividing both sides by π.
Then take the square root of both sides.
or
.
Linear Equation
A linear equation is an equation for a straight line
Example: y = 2x+1 is a linear equation:
The graph of y = 2x+1 is a straight line
When x increases, y increases twice as fast, hence 2x
When x is 0, y is already 1. Hence +1 is also needed
So: y = 2x + 1
Here are some example values:
x y = 2x + 1
-1 y = 2 × (-1) + 1 = -1
0 y = 2 × 0 + 1 = 1
1 y = 2 × 1 + 1 = 3
2 y = 2 × 2 + 1 = 5
Check for yourself that those points are part of the line above!
y=mx+c
m= slope c= y intercept
