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Simultaneous Equations (Cambridge O Level Maths)

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Linear Simultaneous Equations

What are linear simultaneous equations?

  • When there are two unknowns (say x and y) in a problem, we need two equations to be able to find them both: these are called simultaneous equations
    • you solve two equations to find two unknowns, x and y
      • for example, 3x + 2y = 11 and 2x - y = 5
      • the solutions are x = 3 and y = 1
  • If they just have x and y in them (no x2 or y2 or xy etc) then they are linear simultaneous equations

How do I solve linear simultaneous equations by elimination?

  • "Elimination" completely removes one of the variables, x or y
  • To eliminate the x's from 3x + 2y = 11 and 2x - y = 5 
    • Multiply every term in the first equation by 2
      • 6x + 4y = 22
    • Multiply every term in the second equation by 3
      • 6x - 3y = 15
    • Subtract the second result from the first to eliminate the 6x's, leaving 4y - (-3y) = 22 - 15, i.e. 7y = 7
    • Solve to find y (y = 1) then substitute y = 1 back into either original equation to find x (x = 3)
  • Alternatively, to eliminate the y's from 3x + 2y = 11 and 2x - y = 5 
    • Multiply every term in the second equation by 2
      • 4x - 2y = 10
    • Add this result to the first equation to eliminate the 2y's (as 2y + (-2y) = 0)
      • The process then continues as above
  • Check your final solutions satisfy both equations

How do I solve linear simultaneous equations by substitution?

  • "Substitution" means substituting one equation into the other
  • Solve 3x + 2y = 11 and 2x - y = 5 by substitution
    • Rearrange one of the equation into y = ... (or x = ...)
      • For example, the second equation becomes y = 2x - 5 
    • Substitute this into the first equation (replace all y's with 2x - 5 in brackets)
      • 3x + 2(2x - 5) = 11
    • Solve this equation to find x (x = 3), then substitute x = 3 into y = 2x - 5 to find y (y = 1)
  • Check your final solutions satisfy both equations

How do you use graphs to solve linear simultaneous equations?

  • Plot both equations on the same set of axes
    • to do this, you can use a table of values or rearrange into y = mx + c if that helps
  • Find where the lines intersect (cross over)
    • The x and y solutions to the simultaneous equations are the x and y coordinates of the point of intersection
  • e.g. to solve 2x - y = 3 and 3x + y = 7 simultaneously, first plot them both (see graph)
    • find the point of intersection, (2, 1)
    • the solution is x = 2 and y = 1

Solving Equations Graphically Notes Diagram 1, A Level & AS Level Pure Maths Revision Notes

Exam Tip

  • Always check that your final solutions satisfy the original simultaneous equations - you will know immediately if you've got the right solutions or not

Worked example

Solve the simultaneous equations

5x + 2y = 11
4x - 3y = 18

Number the equations.

table row cell 5 x space plus space 2 y space end cell equals cell space 11 space space space space space space space space space space space space end cell row cell 4 x space minus space 3 y space end cell equals cell space 18 end cell end tabletable attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 1 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 2 close parentheses end cell end table 

Make the y terms equal by multiplying all parts of equation (1) by 3 and all parts of equation (2) by 2.
This will give two 6y terms with different signs. The question could also be done by making the x terms equal by multiplying all parts of equation (1) by 4 and all parts of equation (2) by 5, and subtracting the equations.

table row cell 15 x space plus space 6 y space end cell equals cell space 33 space space space space space space space space space end cell row cell 8 x space minus space 6 y space end cell equals cell space 36 space space space space space space space space end cell end tabletable row blank blank cell open parentheses 3 close parentheses end cell end table table row blank blank cell open parentheses 4 close parentheses end cell end table

The 6y terms have different signs, so they can be eliminated by adding equation (4) to equation (3). 

space space space space space space space space space space 15 x space plus space 6 y space equals space 33 space space space space space space space space space space space space bottom enclose space plus open parentheses space space space space space space 8 x space minus space 6 y space equals space 36 close parentheses space end enclose space space space space space space space space space space 23 x space space space space space space space space space space space space space equals space 69 space

Solve the equation to find x  by dividing both sides by 23.

table row cell x space end cell equals cell space 69 over 23 equals space 3 end cell end table

Substitute x space equals space 3 into either of the two original equations.

open parentheses 1 close parentheses space space space space space 5 open parentheses 3 close parentheses space plus space 2 y space equals space 11

Solve this equation to find y.

table attributes columnalign right center left columnspacing 0px end attributes row cell 15 space plus space 2 y space end cell equals cell space 11 end cell row cell 2 y space end cell equals cell space 11 space minus space 15 end cell row cell 2 y space end cell equals cell space minus 4 space end cell row cell y space end cell equals cell fraction numerator negative 4 over denominator 2 end fraction space equals space minus 2 end cell end table

Substitute x = 3  and y = - 2 into the other equation to check that they are correct

table row blank blank cell open parentheses 2 close parentheses space space space space space end cell end table table attributes columnalign right center left columnspacing 0px end attributes row cell 4 x space minus space 3 y space end cell equals cell space 18 end cell end table
   table row cell 4 open parentheses 3 close parentheses space minus space 3 open parentheses negative 2 close parentheses space end cell equals cell space 18 end cell row cell 12 space minus open parentheses negative 6 close parentheses space end cell equals cell space 18 end cell row cell 18 space end cell equals cell space 18 end cell end table

bold italic x bold space bold equals bold space bold 3 bold comma bold space bold space bold italic y bold equals bold minus bold 2

Forming Simultaneous Equations

How do I form simultaneous equations?

  • Introduce two letters, e.g. x and y, to represent the two variables (unknowns)
    • make sure you know exactly what they stand for (and any units)
  • Create two different equations from the words or contexts
  • e.g. 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 and give answers in context (with units)
    • x = 40, y = 30
    • an apple costs 40p and a banana costs 30p
  • Some questions don't explicitly tell you to "solve simultaneously" (even though you need to)
    • e.g. if two numbers have a sum of 19 and a difference of 5, what's their product?
      • x + y = 19 and x - y = 5
      • solve simultaneously to get x = 12, y = 7
      • the question asks for the product, so work out xy = 12 × 7 = 84
  • Check you've answered the question

Worked example

EPS Notes fig2 (1), downloadable IGCSE & GCSE Maths revision notes

EPS Notes fig2 (2), downloadable IGCSE & GCSE Maths revision notes

Exam Tip

  • If you can use the first letters of the unknowns as your variables to help you keep on top of which variable represents which unknown in your working
    • e.g. using a for adults and c for children 
    • this can only work if the unknowns start with different letters!

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    Amber

    Author: Amber

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.