Algebra Basics (Edexcel GCSE Maths: Foundation)

Revision Note

Test yourself

Author

Mark

Last updated

12 March 2024

Substitution

What is substitution?

Substitution means replacing a letter (variable) in a formula with a given number

How do I substitute numbers into a formula?

Write down the formula
Replace (substitute) the letters in the formula with the given numbers
- If substituting in a negative number, it is important to put brackets around it
  - For example, (-3)
Simplify any numerical calculations
Calculate the final value
Sometimes the result is an equation which you can then solve

Examiner Tip

In the calculator paper, use your calculator to work out the final value.
- Don't forget to type out brackets around any substituted negative numbers!

Worked example

(a)

Find the value of the expression

2 x (x + 3 y)

when

x = 2

and

y = - 4

Substitute the numbers given
Use brackets () around negative numbers

$\begin{array}{cl} 2 \times 2 \times (2 + 3 \times (- 4)) \end{array}$

Complete the calculation
Show every step of working, following the order of operations correctly

$\begin{array}{cl} = & 2 \times 2 \times (2 - 12) \\ = & 2 \times 2 \times (- 10) \\ = & 4 \times (- 10) \end{array} \begin{array}{l} \end{array}$

$- 40$

(b)

The formula

P = 2 l + 2 w

is used to find the perimeter,

P

, of a rectangle of length

l

and width

w

Given that the rectangle has a perimeter of 20 cm and a width of 4 cm, find its length.

Substitute the values you are given for $P$ and $w$ into the formula

$20 = 2 \times l + 2 \times 4$

Simplify

$20 = 2 l + 8$

Solve the resulting equation to find the value of $l$
Start by subtracting 8 from both sides

$12 = 2 l$

Divide both sides by 2

$l = 6 cm$

Collecting Like Terms

What happens if there is more than one term?

Terms can be added and subtracted
- The numbers in front of the letters are called coefficients
Each term has a positive or negative sign in front
- In 2x - 3y the sign of the x term is positive and the sign of the y term is negative
Subtractions can be thought of as adding a negative
- 2x - 3y is the same as 2x + (-3y)
  - Just like 20 - 30 is the same as 20 + (-30)
The order of two terms can be swapped, but the signs must move with their terms
- 2x - 3y is the same as -3y + 2x
  - A plus is now needed in front of the 2x
  - Just like 20 - 30 is the same as -30 + 20
If no number appears in front of a term, then its number is 1
- x is the same as 1x

What is a like term?

Like terms are terms with exactly the same letters and powers
- The numbers in front can be different
  - For example:
    2x and 3x
    4x² and 6x²5xy and -7xy
- These are not like terms:
  - 2x and 3y (different letters)
  - 4x² and 6x⁴(different powers)
  - 5xy and 7xyz (different letters)
Remember multiplication can be done in any order
- xy and yx are like terms
  - So are 2xy and 3yx

How do I collect like terms?

Collecting like terms means simplifying by adding or subtracting the numbers in front
- 2x + 3x becomes 5x
- 4y - 10y becomes -6y
  - A negative sign is needed here
If there are different types of like terms, collect them separately
- For 2x + 4y + 5x - 3y
  - Collecting the x's gives 2x + 5x = 7x
  - Collecting the y's gives 4y - 3y = y
  - The answer is 7x + y

Examiner Tip

Don't leave terms like 1x in your final answer in an exam - always simplify them to just x.

Worked example

Simplify

$8 a - 5 b - 6 a + 4 b$

Collect the a terms first

$8 a - 6 a = 2 a$

Then collect the b terms
Don't forget the minus sign in front of the 5b

$- 5 b + 4 b = - b$

Add together the two answers

$2 a + - b$

Simplify the signs

$2 a - b$

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