ALGEBRA

Introduction

Introduction

The language of Algebra

Algebra is a universal language that has straightforward rules and concepts.

Placing algebra in context enables us to see that algebra is all around us in an everyday setting.

For example the letter c could represent the cost of a taxi journey where n represents the number of miles travelled.

The letter x could represent the blood sugar level of an individual with diabetes and y could represent the irregular heart beat of a patient.

Nore: 2 x c is written as 2c in algebra. a b is written as a/b

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Mixed Numbers: Gathering like terms

MathsConsider a classroom of 30 students as follows:

Front row – 7 males and 3 females

Second row – 4 males and 6 females

Back row – 5 males and 5 females

If we let f = female and m = male we have:

7m + 3f + 4m + 6f + 5m + 5f

which simplifies to 16m + 14f.

This final line is the process of ‘gathering like terms together’

Gathering like terms together is a straightforward process which is often facilitated by imagining each ‘family’ of letters in a specific colour. We can think of x terms in red, y terms in blue and z terms in green

For example, simplify 3x + 4y + 5z + 9x – 12y + 7z

Think red: 3x + 9x = 12x

Think blue: 4y – 12y = -8y (a vision of the number line helps here)

Think green: 5z + 7z = 12z

Answer : 12x – 8y + 12z

If we have one x or one y or one z we should write this simply as x rather than 1x, y rather than 1y and z rather than 1z.

When gathering like terms together we must be careful not to mix terms together:

4a + 3b + 2ab + 5a +2b + 9ab

= 9a + 5b + 11ab

We cannot combine 9a and 5b together in any way.

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Inequalities

You can recognise an inequality because it is two expressions that are joined by one of the following symbols:

a ? b means a is less than b

a ? b means a is more than b

a ? b means a is less than or equal to b

a ? b means a is more than or equal to b

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Number lines and graphs

It’s possible to view an inequality visually as a number line.

To represent ? or ? you use an open circle.

To represent ? or ? you use a solid circle

Maths

You can also show inequalities using a graph, in which case the inequalities are shown in the shaded region of the graph.

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More than one inequality (higher tier)

For the higher tier you should know about using more than one inequality.

For example: what integer values of b satisfy the inequalities -2 ? b ? 7 and b ? 0?

If -2 ? b ? 7 then b = -2, -1, 0, 1, 2, 3, 4, 5, 6,

If b ? 0 then b = any number larger than 0

the values that satisfy both equations are 1, 2, 3, 4, 5 and 6.

This means the solution is:

1 ? b ? 6

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Multiplying out brackets

If given an equation with brackets, you may need to multiply the brackets out. To do this, simply multiply the term outside the brackets by each term within the brackets. For example, if we have

5(x + 4)

we multiply the x by 5, and the 4 by 5, to give us

5x + 20

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Multiplying out two brackets (higher tier)

For the higher tier you’ll be expected to know how to multiply out two brackets.

Basically, all the terms in the second set of brackets need to be multiplied by all the terms in the first set of brackets.

For example: multiply out (x + 2) (x + 5)

First of all multiply the second bracket by x:

x2 + 5x

Then you multiply the second bracket by 2:

2x + 10

And your final answer is:

x2 + 5x + 2x + 10

Which can be simplified to:

x2 + 7x + 10

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Brackets and power

The important point to remember about powers is that the power amount represents the number of times the bracket multiplies by itself. So, if you’re asked to multiply out a bracket which is squared, write it out completely first.

For example: multiply out (b + 4)2

First of all write out the expression like this:

(b + 4) (b + 4)

Now you can expand out the brackets as before:

(b + 4) (b + 4)

= b2 + 4b + 4b + 16

= b2 + 8b + 16

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Factorising

Factorising is the opposite of multiplying out brackets. To factorise an expression, you look for the highest common factor of the terms, and bring it outisde the brackets. For example, in the expression

5a + 15

both numbers have a common factor of 5. The first term can be expressed as 5 x a, and the second term as 5 x 3. We can theremore bring 5 outside the brackets and write

5 (x + 3)

Here is another example:

14y + 21

The common factor here is 7, and we can write this as

7 (2y + 3)

If all the terms contain an x, then we can bring the x outside the brackets aswell, like in the following examples

4x2 + 12x = 4x (x + 3)

12x3 + 9x2 = 3x2 (4x + 3)

2y3 + 4y2 + 6y = 2y(y2 + 2y + 3)

REMEMBER:

x2 x x = x3

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