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Algebra Basics (AQA GCSE Maths)

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Paul

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Paul

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Substitution

What is substitution?

  • Substitution is where we replace letters in a formula with their values
  • This allows you to find one other value that is in the formula

 

How do we substitute numbers into a formula?

  • Write down the formula, if not clearly stated in question
  • Substitute the numbers given, using brackets around negative numbers, (-3), (-5) etc
  • Simplify any calculations if you can
  • Rearrange the formula if necessary (it is usually easier to substitute first)
  • Work out the calculation (use a calculator if allowed)

 

Are there any common formulae to be aware of?

  • Formulae for accelerating objects are often used
    • v equals u plus a t
    • v squared equals u squared plus 2 a s
    • s equals u t plus 1 half a t squared
  • The letters mean the following:
    • t stands for the amount of time something accelerates for (in seconds)
    • u stands for its initial speed (in m/s) - the speed at the beginning
    • v stands for its final speed (in m/s) - the speed after t seconds
    • a stands for its acceleration (in m/s2) during in that time
    • s stands for the distance covered in t seconds
  • You do not need to memorise these formulae, but you should know how to substitute numbers into them
    • You should also understand the difference between initial speed (u) and final speed (v)

Worked example

(a)
Find the value of the expression 2 x open parentheses x plus 3 y close parentheses when x equals 2 and y equals negative 4.
 

Substitute the numbers given.
Use brackets () around negative numbers that you have substituted so that you don't forget about them.
It is a good idea to show every step of working to make sure that you are following the order of operations correctly.

table row blank blank cell 2 cross times 2 cross times open parentheses 2 plus 3 cross times open parentheses negative 4 close parentheses close parentheses end cell row blank equals cell 2 cross times 2 cross times open parentheses 2 minus 12 close parentheses end cell row blank equals cell 2 cross times 2 cross times open parentheses negative 10 close parentheses end cell row blank equals cell 4 cross times negative 10 end cell end table

bold minus bold 40

  

(b)
The formula P equals 2 l plus 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 into the formula.

20 equals 2 cross times l plus 2 cross times 4

Simplify.

20 equals 2 l plus 8

Subtract 8 from both sides.

12 equals 2 l

Divide both sides by 2.

bold italic l bold equals bold 6 bold space bold cm

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Collecting Like Terms

How do we collect like terms?

  • TERMS are separated by + or –
    • The sign belongs to the coefficient of the term after the symbol
    • If there is no symbol in front of the first term then this is a positive term
      • 2x - 3y means +2 x's and -3 y's
  • “LIKE” terms must have exactly the same LETTERS AND POWERS (the COEFFICIENT can be different)
    • Examples of like terms: 
      • 2x and 3x
      • 2x2 and 3x2
      • 2xy and 3xy
      • 4(x + y) and 5(x + y)
    • Examples that are NOT like terms
      • 2x and 3y (different letters)
      • 2x2 and 3x4 (different powers)
      • 2xy and 3xyz (different letters)
      • 4(x + y) and 5(x + y)2 (different powers)
    • Remember multiplication can be done in any order
      • xy and yx are like terms
  • Add the COEFFICIENTS of like terms
    • If the answer is a positive answer then put "+" in front if there are other terms before it
      • x - 2y + 5y = x + 3y
    • If the answer is a negative number then put "-" in front
      • x - 5y + 2y = x - 3y

Examiner Tip

  • A “coefficient” answers the question “how many?”

   For example:

   the coefficient of x in 2x2 – 5x + 2 is -5

   and:

   the coefficient of x in ax2 + bx + c is b

Worked example

Simplify  x squared minus 3 x y plus 2 x squared plus 4 x minus 2 x y minus x plus 7.

Reorder the terms so that the x squared's are together, as are the x y's, x's and the constants.
Make sure that you keep the same sign in front of them when you reorder them.

x squared plus 2 x squared minus 3 x y minus 2 x y plus 4 x minus x plus 7

Add the coefficients of the 'like' terms.

bold 3 bold italic x to the power of bold 2 bold minus bold 5 bold italic x bold italic y bold plus bold 3 bold italic x bold plus bold 7

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    Paul

    Author: Paul

    Expertise: Maths

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.