ALGEBRA

In algebra, a letter can be used to represent a number that you do not know. This gives us algebraic terms, like 2x and 5y.

When algebraic terms are put together by mathematical operations such as + and - we get an algebraic expression, like 2x + 5y +3z.

We can simplify algebraic expressions by collecting 'like terms'.

Days of the week

If you are asked how many days there are in a week, you would find the answer easy - there are 7. How about the number of days in two weeks? Again, the answer is fairly straightforward - there are 14.

What happens when you are asked how many days there are in three weeks, or four or five? If you know the formula you could work this out.

1 week is 1 × 7 days = 7
2 weeks is 2 × 7 days = 14
3 weeks is 3 × 7 days = 21
4 weeks is 4 × 7 days = 28

So, the formula to work out the number of days is:
number of days = number of weeks × 7

In algebra, letters are used when numbers are not known.

Algebraic terms, like 2s or 8y, leave the multiplication signs out. So rather than '2 × s', write 2s, rather than '8 × y' write 8y.

A string of numbers and letters joined together by mathematical operations such as + and - is called an algebraic expression.

r + 2s means an unknown number 'r', plus 2 lots of an unknown number s

Test youself questions

Q1. Say that 'g' is the cost of child admission, and 'k' is the cost of adult admission to the zoo.

a) How much does it cost for the Khan family of 3 children and 3 adults to visit the zoo?

 b) Write an algebraic expression for the cost for the Norman family of 5 children and 4 adults to visit the zoo.

Like terms

Algebraic terms that have the same letter are called like terms. Only like terms can be added or subtracted.

Example

So 9b, -7b and 13b are like terms, but 6t, 5x and -11z are not like terms.

When like terms are added and subtracted it is called simplifying.

Adding and subtracting like terms

The Khan family and the Norman family visit the zoo together - there are 8 children and 7 adults in the group. Because there are more than 10 people, the families can take advantage of a special offer - 1 child can be admitted free of charge.

As before let's use g for the cost of child admission, and k for the cost of adult admission.

Cost for the Khan family = 3g + 3k

Cost for the Norman family = 5g + 4k

Offer = - g

Total cost = 3g + 3k + 5g + 4k - g

Simplified = 3g + 5g - g + 3k + 4k = 7g + 7k

+ and - signs

A term, like a number, belongs with the sign that sits in front of it. So in the expression 2k - g, g belongs with the - sign that sits in front of it, so it is - g; and 2k belongs with the + sign that sits in front of it, making it + 2k.

Remember:

• Collect all the like terms together, eg, re-write the expression 3g + 2k + 5g + 4k - g with all the g s and all the k s together: 3g + 5g - g + 2k + 4k
• When you add or subtract terms, keep each term with their + or - sign.

 Simplify 4x + y - 2x + 6y.                                                                                                                           Re-write the expression by putting the like terms next to each other. remember to keep the plus or minus signs with the terms they belong to.
4x + y - 2x + 6y
= 4x - 2x + y + 6y
Simplify this to get:
2x + 7y.

Like terms with powers can be added or subtracted but only if the powers are the same.

x is not the same as x², so they cannot be added together.

Example

Simplify:
3x + 5 + x² - 2x + 2x² - 1

Solution

Rearrange the expression so that like terms are next to each other:
x² + 2x² + 3x - 2x + 5 - 1

And then simplify:
3x² + x + 4

When multiplying expressions in brackets, make sure that everything inside the bracket is multiplied by the term (or number) outside the bracket.

Example

Expand 2(3x + 4)

2 x 3x = 6x

2 x 4 = 8

2(3x + 4)= 6x + 8

Bracket x bracket

What happens when we have more than a single term or number outside the bracket? What happens when we have another bracket.

For example, if we want to expand (a + b)(c + d), we need to make sure that everything in the second bracket is multiplied by everything in the first.

first we:

a x c=ac, a x d= ad, b x c=bc, b x d=bd

(a+b)(c+d)= ac+ad+bc+bd

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.

Equations are made up of two expressions on either side of an equals sign, like
x + 2 = 1
To solve an equation, you need to find the values of the missing numbers.

Example

'I think of a number, add four, and the answer is seven.'

Written algebraically, this statement is 'x + 4 = 7', where 'x' represents the number you thought of.

'x + 4 = 7' is an example of an algebraic equation. 'x' represents an unknown number.

The number you first thought of must be three (3 + 4 = 7). Therefore, x = 3 is the solution to the equation x + 4 = 7.

An equation is like a weighing scale - both sides should always be perfectly balanced. To solve the equation you need to find the value of missing numbers and perform the same operation to each side.

Example 1

For example, suppose you are trying to find out how many sweets are in the bag shown here. By subtracting three sweets from each side, the scales remain balanced.

You can now see that one bag is equivalent to two sweets. Written algebraically, this is:

x + 3 = 5
Subtract 3 from both sides, to give:
x = 2

Example 2

In this case, two bags of sweets are equivalent to six sweets.

To find the equivalent of one bag, divide both sides in half:

Written algebraically, this is:

2x = 6
Divide both sides by 2, to give:
x = 3

test yourself

Solve the equation:                                                                                                  a) a - 3 = 4
b) 5b = 35

Sometimes an equation will have multiples of an unknown, eg, 5y = 20. To solve this you need to get the unknown on its own. To do this, divide both sides by 5.

5y = 20
5y ÷ 5 = 20 ÷ 5
y = 4

Sometimes an equation will have multiples of an unkown and other numbers, eg, 3x + 2 = 8

In equations of this type, your aim is to get all the 'x's (or unknowns) on one side and all the numbers on the other.

Example

Let's solve the equation 3x + 2 = 8

3x + 2 = 8

We want to get the 'x' on it's own. Start by subtracting 2 from both sides:

3x + 2 - 2 = 8 - 2
3x = 6

Then divide by 3:

So x = 2

The same rules apply if an equation has x on both sides. Keep the equation balanced and perform the same operation to both sides.

Example

Solve the equation 2x + 2 = x + 4 Aim to get unknowns on just one side of the equation, so begin by subtracting x from each side

Now you have the type of equation that you recognise, all you need to do is subtract 2 from both sides.

Written algebraically, this becomes:

2x + 2 = x + 4
Subtract x from both sides to give x + 2 = 4
Subtract 2 from both sides to give x = 2

If the value of x is negative, tackle the equation in the same way - aim to get all xs on one side of the equation.

Example 1

Solve the equation 5x - 2 = 12 - 2x. Your aim is to get all the unknown x terms on one of the equation side only, so start by adding 2x to both sides: 7x - 2 = 12
Next add 2 to both sides: 7x = 14
And finally, divide by 7 to give x = 2

As always, you can check your answer in the original equation. So substitute x = 2 back into 5x - 2 = 12 - 2x

(5 × 2) - 2 = 12 - (2 × 2)
10 - 2 = 12 - 4
8 = 8
This makes sense, so the value x = 2 is correct.

In an equation the '=' sign means that the two sides are identical. But what happens when the two sides are not identical?

If this is the case you need to use inequalities to show the relationship between the two sides.

< means 'less than'
≤ means 'less than or equal to'
> means 'greater than'
≥ means 'greater than or equal to'

For example, if x > 2, then x = 3, 4, 5, 6, 7, ... (x is greater than, but not equal to 2, so don't include the 2).

If y is ≤ 4, then y = 4, 3, 2, 1, 0, -1, ... (y is less than or equal to 4, so do include the 4).

All graphs have an x-axis and a y-axis.

The point (0,0) is called the origin, The horizontal axis is the x-axis, The vertical axis is the y-axis

Coordinates

Coordinates are written as two numbers, separated by a comma and contained within round brackets. For example, (2, 3), (5, 7) and (4, 4)

• The first number refers to the x coordinate.

• The second number refers to the y coordinate.

Coordinates are written alphabetically - so x comes before y (x, y). One way to remember is 'you go along the hallway before you go up the stairs'

Extending the x and y axes beyond the origin reveals the negative scales. The areas of the graph between axes are called quadrants. So now we have four quadrants in total.

Coordinates in these quadrants are still described in terms of x and y. But now we can have negative values for x, y or both.

Graphs of this type have either a horizontal or vertical straight line.

Example y = 4

All the points on this line have a y coordinate of 4 so you can say that the equation of the line is y = 4.

Example x = 7                                                                                                    All the points on this line have an x coordinate of 7 so you can say that the equation of the line is x = 7.

Straight lines

The equation of a straight line on a graph is made up of a y term, an x term, and a number and are written in the form of y = mx + c.

• The slope of the line is known as the gradient and is represented by m in the equation.

• The point at which the line crosses the y-axis is the c in the equation.

All the points that lie on the blue line have a y coordinate that is the same as the x coordinate
eg (-1,-1) and (2,2)
We say that the equation of the line is y = x

All the points that lie on the orange line have a y coordinate (the second number in brackets) that is one number higher than the x coordinate of the same line eg
(-3,-2) and (0,1)
In other words, the y coordinate equals the x coordinate + 1
So the equation of the line is y = x + 1