National 5MathsSQARevision NotesAlgebraic SkillsSimultaneous EquationsConstruction of Simultaneous Equations

Construction of Simultaneous Equations (SQA National 5 Maths): Revision Note

Exam code: X847 75

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

Constructing simultaneous equations from text

How do I form simultaneous equations from text descriptions?

  • Introduce two letters, for example 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 pence
        y is the price of a banana, in pence

      • This question could also be done in pounds, £

  • Solve the equations simultaneously

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

    • x = 40, y = 30

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

  • 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.

Examiner Tips and Tricks

You don't need to use x and y as your variables.

If an exam question asks you to use a particular set of letters, be sure to follow those instructions.

Worked Example

At a bakery, Eilidh pays £9 in total for six bagels and twelve chocolate bites.

Let b be the price of a bagel and c be the price of a chocolate bite.

(a) Write down an equation in b and c to illustrate this information.
 

At the same bakery, Iain buys nine bagels and ten chocolate bites.

His total cost is £12.30.

(b) Write down an equation in b and c to illustrate this information.
 

Bevan buys 5 bagels and 15 chocolate bites from the bakery.

(c) Calculate the total cost of Bevan's purchase.
 

Answer:

Part (a)

The two variables are the price of a bagel, b, and the price of a chocolate bite, c

Write an equation for the Eilidh's purchase, using prices in pounds

  • You could use prices in pence instead

  • In that case you would have to convert £9 to 900p for the right hand side of the equation

6 b plus 12 c equals 9
 

Part (b)

Write an equation for Iain's purchase, using prices in pounds

  • You could use prices in pence instead

  • In that case you would have to convert £12.30 to 1230p for the right hand side of the equation

  • Make sure you use the same units for both equations!

9 b plus 10 c equals 12.3
 

Part (c)

Label the equations 1 and 2

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

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

'Rescale' the equations to eliminate the b terms

  • You could also choose to eliminate the c terms first

  • In that case you would multiply equation 1 by 5 and equation 2 by 6 before subtracting the equations

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 c 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 c 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 space space space space 18 b plus 36 c equals 27 space space space space space space space space space space space space bottom enclose negative open parentheses 18 b plus 20 c equals 24.6 close parentheses space end enclose space space space space space space space space space space space space space space space space 16 c equals 2.4 space

Solve for c

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

Substitute this into either equation to find b

  • Here 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 chocolate bites cost £0.15 each and bagels cost £1.20 each

Use these values to find the price of 5 bagels and 15 chocolate bites

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

£8.25

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject 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.