Algebra Basics (Edexcel GCSE Maths: Foundation)

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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 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

table row 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 end table

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

=2×2×2-12=2×2×-10=4×-10{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}

-40{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}

  

(b)
The formula P=2l+2w{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}  is used to find the perimeter, P{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true},  of a rectangle of length l{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}  and width w{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}.
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{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}  and w{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true} into the formula

20=2×l+2×4{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}

Simplify

20=2l+8{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}

Solve the resulting equation to find the value of l{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}
Start by subtracting 8 from both sides

12=2l{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}

Divide both sides by 2

l=6 cm{"language":"en","fontFamily":"Times New Roman","fontSize":"18","autoformat":true}

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 - 3the sign of theterm is positive and the sign of the 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
    • 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 6x2
        5xy  and -7xy
    • These are not like terms:
      • 2x  and 3(different letters)
      • 4x2  and 6x(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 - 10 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 minus 5 b minus 6 a plus 4 b

 

Collect the a terms first

8 a minus 6 a equals 2 a
 

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

negative 5 b plus 4 b equals negative b
 

Add together the two answers

2 a plus negative b

 

Simplify the signs

bold 2 bold italic a bold minus bold italic b

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Mark

Author: Mark

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.