Revision NotesPast Papers
IGCSEMathsCambridge (CIE)International MathsCoreRevision NotesAlgebra & SequencesSimultaneous EquationsForming Simultaneous Equations

Forming Simultaneous Equations (Cambridge (CIE) IGCSE International Maths)

Revision Note

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Cambridge (CIE) International Maths: CoreCambridge (CIE) Maths: Core

Forming Simultaneous Equations

How do I form simultaneous equations?

  • Introduce two letters, x and y, to represent two unknowns

    • Make sure you know exactly what they stand for (and any units)

  • Create two different equations from the words or contexts

    • 3 apples and 2 bananas cost $1.80, while 5 apples and 1 banana cost $2.30 

      • 3x + 2y = 180 and 5x + y = 230
        x is the price of an apple, in cents
        y is the price of a banana, in cents

      • This question could also be done in dollars, $

  • Solve the equations simultaneously

  • Give answers in context (relate them to the story, with units)

    • x = 40, y = 30

    • In context: an apple costs 40 cents and a banana costs 30 cents

  • Some questions don't ask you to solve simultaneously, but you still need to

    • Two numbers have a sum of 19 and a difference of 5, what is their product?

      • x + y = 19 and x - y = 5

      • Solve simultaneously to get x = 12, y = 7

      • The product is xy = 12 × 7 = 84

Examiner Tips and Tricks

  • Always check that you've answered the question! Sometimes finding x and y isn't the end

    • E.g. you may have to state a conclusion

Worked Example

At a bakery a customer pays £9 in total for six bagels and twelve sausage rolls.

Another customer buys nine bagels and ten sausage rolls, which costs £12.30 in total.

Find the cost of 5 bagels and 15 sausage rolls.

The two variables are the price of bagels, b, and the price of sausage rolls, s

Write an equation for the first customer's purchases, and label it equation 1

circle enclose 1 space space space 6 b plus 12 s equals 9

Write an equation for the second customer's purchases, and label it equation 2

circle enclose 2 space space space 9 b plus 10 s equals 12.3

We will choose to eliminate the b terms
Make the b terms equal by multiplying all parts of equation 1 by 3 and all parts of equation 2 by 2
Label these as equations 3 and 4

table row cell circle enclose 1 cross times 3 space space space 18 b plus 36 s end cell equals cell 27 space space space space space space space space space space circle enclose 3 end cell row cell circle enclose 2 cross times 2 space space space 18 b plus 20 s end cell equals cell 24.6 space space space space space space space circle enclose 4 end cell end table

To eliminate b, subtract equation 4 from equation 3

circle enclose 3 minus circle enclose 4 space space space space space 16 s equals 2.4

Solve for s

s equals fraction numerator 2.4 over denominator 16 end fraction equals 0.15

Substitute this into either equation to find b, we will use equation 1

table row cell circle enclose 1 space space space 6 b plus 12 open parentheses 0.15 close parentheses end cell equals 9 row cell 6 b plus 1.8 end cell equals 9 row cell 6 b end cell equals cell 7.2 end cell row b equals cell 1.2 end cell end table

So sausage rolls cost £0.15 each and bagels cost £1.20 each

Use these values to find the price of 5 bagels and 15 sausage rolls

open parentheses 5 cross times 1.2 close parentheses plus open parentheses 15 cross times 0.15 close parentheses equals 8.25

£8.25

Last updated:

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

the (exam) results speak for themselves:

    Did this page help you?

    Previous:Solving Simultaneous EquationsNext:Introduction to Sequences
    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    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.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.