Linear Simultaneous Equations (CIE IGCSE Additional Maths)

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Amber

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Elimination Method

What are simultaneous linear 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
  • Solving a pair of simultaneous equations means finding pairs of values that make both equations true at the same time
  • A linear equation in two unknowns will produce a straight line if you graph it... linear = line
  • A pair of simultaneous equations will produce lines that will cross each other (if there is a solution!)

How do I use elimination to solve simultaneous linear equations?

  • "Elimination" completely removes one of the variables, x or y
  • Begin by multiplying one (or both) of the equations by a constant (or constants)to get the numbers in front of one of the unknowns to match
    • For example 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
  • If the matching numbers have the same sign, then subtract one equation from the other
  • If the matching numbers have different signs then add the equations together
    • Subtract the second result from the first to eliminate the 6x's
      • 4y - (-3y) = 22 - 15
      • 7y = 7
  • Solve the new equation to find the value of one of the unknowns
    • Solve to find y
      • y = 1
  • Substitute the value into one of the original equations and solve to find the value of the other unknown
    • Substitute y = 1 back into either original equation
      • 3x + 2(1) = 11
      • 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
    • 3(3) + 2(1) = 11 and 2(3) - (1) = 5

How do I solve linear simultaneous equations from worded contexts?

solving linear simultaneous equations from worded contexts part 1

solving linear simultaneous equations from worded contexts part 2

Examiner Tip

  • Don't skip the checking step (it only takes a few seconds) – there are many places to go wrong when solving simultaneous equations!
  • Mishandling minus signs is probably the single biggest cause of student error in simultaneous equations questions

Worked example

solving linear simultaneous equations worked example

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Substitution Method

How do I use substitution to solve simultaneous linear equations?

  • "Substitution" means substituting one equation into the other
  • Rearrange one of the equations to make one of the unknowns the subject 
    • To 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 the expression found for or y into the equation not used to rearrange
    • Replace all y's with 2x - 5 in brackets
      • 3x + 2(2x - 5) = 11
  • Solve the new equation to find the value of one of the unknowns
    • Solve this equation to find x
      • x = 3
  • Substitute the value found for or y into the rearranged equation from to find the value of the other unknown
    • Substitute x = 3 into y = 2x - 5 to find y
      • y = 2(3) - 5
      • y = 1
  • Check your final solutions satisfy both equations
      • 3(3) + 2(1) = 11 and 2(3) - (1) = 5

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 + 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 = 4 simultaneously, first plot them both (see graph)
    • find the point of intersection, (2, 1)
    • the solution is x = 2 and y = 1

Intersection of two lines as the solution to a pair of simultaneous equations

Examiner Tip

  • Although elimination will always work to solve simultaneous linear equations, sometimes substitution can be easier and quicker
  • Knowing both methods can help you a lot in the exam (plus you will need substitution to solve quadratic simultaneous equations)

Worked example

worked example for solving simultaneous equations by substitution

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Amber

Author: Amber

Expertise: Maths

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.