- When writing expressions in algebra (as opposed to sums in numbers) there are conventions and symbols that are used that take on a particular meaning
- This is what we mean by algebraic notation
- In number work, for adding and subtracting, we use + and –
- In algebra, we still do!
- Examples:
*a*+*b**c*+*d*–*e* - However for multiplication, no symbol is used, and for division, fractions are used
- Examples:
*ab*(means*a*×*b*)

(means*a*÷*b*)

3*ab*(means 3 ×*a*×*b*) - We can have combinations of these of course
- Examples:

(means*a*×*b*+*c*÷3)

The order of operations still apply in algebra so*a*×*b*and*c*÷*d*would happen before the addition - Powers (indices) and roots are used the same way as with numbers
- Examples:
*a*^{2}means*a*×*a*4*a*^{2}means 4 ×*a*^{2}

With the order of operations,*a*^{2}will happen before multiplying by 4 - Brackets also work in the same way as with numbers
- Examples

means but with brackets taking priority, when known, would be worked out first, then we would multiply by 3

means , again with the brackets worked out first

- Algebraic expressions are used in many parts of the course
- You need to be able to understand their meaning and work with them
- e.g. for rearranging a formula

- You may be given a situation in words that you then have to write in algebra
- Which could lead to an equation you may then have to solve

- You need to be able to understand their meaning and work with them

At the start of a competition, Raheem has *p* conkers and Howard has 2 *q* conkers.

At the end of the competition Raheem still has the same number of conkers he started with but Howard won 6 and lost none.

(a)

Write down an expression for the number of conkers that Howard has at the end of the competition.

**
Howard has 2 q + 6
**

At the end of the competition, Raheem and Howard have a total of 40 conkers.

(b)

Write an equation in terms of *p* and *q* that shows the number of conkers Raheem and Howard have in total at the end of the competition.

**
p + 2 q + 6 = 40
**

This can be simplified to *p* + 2 *q* = 34

You need to know the meanings of the word **term** and **factor**, as they are the basic building blocks in algebra.

You need to know the differences between an **expression**, **equation**, **formula**, **identity **and **inequality **in order to fully understand algebra and proofs. You may be asked to identify which is which.

- A
**term**is either… - …a letter (
**variable**) on its own, e.g.*x* - ..a number on its own, e.g. 20
- …a number
**multiplied**by a letter, e.g. 5*x* - The number in front of a letter is called a
**coefficient** - The coefficient of
*x*in the term 6*x*is 6 - The coefficient of
*y*in the term -5*y*is -5 - Terms that are just numbers (with no letters) are called
**constants** - Terms can include powers and more than one letter
- E.g. 6
*xy*, 4*x*^{2},*ab*^{3}*c*, …

** **

- A
**factor**is any number or letter that**divides**a term exactly (with no remainder) - E.g. all the factors of 4
*xy*are 1, 2, 4,*x*, 2*x*, 4*x*,*y*, 2*y*, 4*y*,*xy*, 2*xy*and 4*xy* - A term can be separated into
**factors**that**multiply**together to give that term - E.g. two factors of 5
*x*are 5 and*x* - To
**factorise**means to write something as a multiplication of factors - A
**common factor**is one that divides both terms - E.g. the common factors of 6
*xy*and 4*x*are 2,*x*and 2*x* - The
**highest**(or greatest) common factor is 2*x*

** **

- An
**expression**is an algebraic statement that does**not**have an**equals sign** - There is nothing to solve
- An expression is made by adding, subtracting, multiplying or dividing
**terms** - E.g. 2
*x*+ 5*y*,*b*^{2}– 2*cd*, , … - A single term can be an expression
- Expressions can be
**simplified**(made easier) - E.g.
*x*+*x*+*x*simplifies to 3*x*

** **

- An
**equation**is an algebraic statement with an**equals sign**between a left-hand side and a right-hand side - Both sides are equal in value
- E.g. if 2
*x*has the same value as 10, then 2*x*= 10 - An equation can be
**solved**by finding the missing values of the letters that make the left-hand side equal to the right-hand side - E.g. the equation 2
*x*= 10 is solved by*x*= 5 *x*= 5 is called the**solution**

** **

- A
**formula**is a worded rule, definition or relationship between different quantities, written in shorthand using**letters** - E.g. weight,
*w*, is mass,*m*, multiplied by gravitational acceleration,*g* - The formula is
*w*=*mg* - It is common to
**substitute**numbers into a formula, but a formula on its own cannot be solved - To turn a formula into an equation, more information is needed
- E.g. In the formula
*w*=*mg*, if*w*= 50 and*m*= 5 then the equation 50 = 5*g*can be formed

** **

- An
**identity**is an algebraic statement with an identity sign, ≡, between a left-hand side and a right-hand side that is**true**for**all values of***x* - E.g.
*x*+*x*≡ 2*x* - This means
*x*+*x*is**identical**to 2*x*, or that*x*+*x*can also be written as 2*x* - An identity
**cannot**be**solved** **All numbers**can be**substituted**into an identity and it will remain true- E.g.
*x*+*x*≡ 2*x*is true for*x*= 1,*x*= 2,*x*= 3 … (even*x*= -0.01,*x*= π etc) - Unlike with equations, where only the solutions work
- E.g. 2
*x*= 10 is not true for*x*= 1,*x*= 2,*x*= 3 … only for*x*= 5 - Identities can be used to
**write**algebraic expressions**in different forms** - E.g. find
*p*and*q*if 3(*x*+*y*) + 2*y*≡*px*+*qy* - 3(
*x*+*y*) expands to 3*x*+ 3*y* - The coefficient of
*x*on the left is 3 and on the right is*p*, so*p*= 3 - The coefficient of
*y*on the left is 3 + 2 and on the right is*q*, so*q*= 5 - Therefore 3(
*x*+*y*) + 2*y*is identical to 3*x*+ 5*y* - This method is called
**equating coefficients**

- An
**inequality**compares a left-hand side to a right-hand side and states which one is bigger *x*>*y*means*x*is greater than*y**x*≥ y means*x*is greater than, or equal to,*y**x*< y means*x*is less than*y**x*≤ y means*x*is less than, or equal to,*y*- E.g.
*x*≥ 8 means*x*can take any value that is greater than, or equal to, 8 - This is the same as saying “8 or more”, or "at least 8"
- The solutions of inequalities are usually, themselves, inequalities
*x*+ 10 < 15 solves to give x < 5, so*x*is any number less than 5

- To fully understand the wording of an exam question you need to know the difference between an expression, equation, formula, identity and inequality.

(a)

Write down the expression from the list below:

2*x* + 5 = 4 3*x* + 2*x* ≡ 5*x * 7*x *– 9 *x *= *vt* – *w* 4*x* – 1 ≥ 0

An expression does not have an equals, identity or inequality sign

**7 x – 9 is the expression**

(b)

For how many values of *x* is the statement *x*^{2} – 1 ≡ (*x* + 1)(*x* – 1) true?

no values of *x* two values of *x* (*x* = 1 and *x* = -1) all values of *x*

* *

This is an identity (due to the ≡ symbol)

An identity is true for all values of*x*

An identity is true for all values of

**all values of x**

(c)

Find the whole numbers *a* and *b *such that 5(*x* – 2*y*) + *ax* + 3*y * ≡ 9*x *+ *by*.

Expand the brackets (by multiplying* x* and -2*y* by 5)

5*x* – 10*y* + *ax* + 3*y * ≡ 9*x *+ *by *

Both sides must be identical

Equate the coefficients of *x* (by setting the number of *x*’s on the left-hand side equal to the number of *x*’s on the right-hand side)

5 + *a* = 9

Solve this equation (by subtracting 5 from both sides)

*a* = 9 – 5

*a*** = 4**

Equate the coefficients of *y* (by setting the number of *y*’s on the left-hand side equal to the number of *y*’s on the right-hand side)

-10 + 3 = *b *

Solve this equation (by adding 3 to -10)

*b*** = -7**

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