Linear Simultaneous Equations (Cambridge (CIE) IGCSE International Maths): Revision Note

Linear simultaneous equations

What are linear simultaneous equations?

• When there are two unknowns (x and y), we need two equations to find them both

  • For example, 3x + 2y = 11 and 2x - y = 5

    • The values that work are x = 3 and y = 1

• These are called linear simultaneous equations

  • Linear because there are no terms like x² or y² 

How do I solve linear simultaneous equations by elimination?

• Elimination removes one of the variables, x or y

• To eliminate the x's from 3x + 2y = 11 and 2x - y = 5, make the number in front of the x (the coefficient) in both equations the same (the sign may be different)

  • Multiply every term in the first equation by 2

    • 6x + 4y = 22

  • Multiply every term in the second equation by 3

    • 6x - 3y = 15

  • Subtracting the second equation from the first eliminates x

    • When the sign in front of the term you want to eliminate is the same, subtract the equations

• The y terms have become 4y - (-3y) = 7y (be careful with negatives) 

  • Solve the resulting equation to find y

  • y = 1

• Then substitute y  = 1 into one of the original equations to find x

  • 3x + 2 = 11, so 3x = 9, giving x = 3

• Write out both solutions together, x = 3 and y = 1

• Alternatively, you could have eliminated the y's from 3x + 2y = 11 and
  2x - y = 5 by making the coefficient of y in both equations the same 

  • Multiply every term in the second equation by 2

  • Adding this to the first equation eliminates y (and so on)

    • When the sign in front of the term you want to eliminate is different, add the equations

How do I solve linear simultaneous equations by substitution?

• Substitution means substituting one equation into the other

  • This is an alternative method to elimination

    • You can still use elimination if you prefer

• To solve 3x + 2y = 11 and 2x - y = 5 by substitution

  • Rearrange one of the equations into y = ... (or x = ...)

    • For example, the second equation becomes y = 2x - 5 

  • Substitute this into the first equation

    • This means 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

How do I solve linear simultaneous equations graphically?

• 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

• For example, to solve 2x - y = 3 and 3x + y = 4 simultaneously

  • First plot them both on the same axes (see graph)

  • Find the point of intersection, (2, 1)

  • The solution is x = 2 and y = 1

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