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Simultaneous Linear Equations

  • In an equation, such as y = 3x – 4, the unknowns x and y are called variables.
  • Simultaneous linear equations can be solved algebraically; either by substitution or by elimination.
  • Simultaneous linear equations can also be solved graphically by drawing the graphs of the two equations on the same axes; their point of intersection gives the solution.
Examples
  1. Solve the equations:
    3a + b = 10
    2a + 4b = 0

    • 3a + b = 10       – (i)
      2a + 4b = 0       – (ii)
    • from (i), b = 10 – 3a       – (iii)
    • Substitute (10 – 3a) for b in (ii): 2a + 4(10 – 3a) = 0
    • Clear the brackets: 2a + 40 – 12a = 0
    • Collect like terms: 2a – 12a = -40
    •                                    -10a = -40
    • ∴ a = 4
    • Substitue a = 4 in (iii): b = 10 – (3 × 4)
    •                                         b = 10 – 12
    • ∴ b = -2
    • Therefore, a = 4, and b = -2
  2.  Solve the equations
    3f = 4 – 4e
    2e = 5f + 15

    • 3f = 4 – 4e        – (i)
      2e = 5f + 15     – (ii)
    • Arrange the equations so that the unknowns are in alphabetical order on the LHS and the numbers are on the RHS:
    • 4e + 3f = 4       – (iii)
      2e – 5f = 15      – (iv)
    • Multiply (iv) by 2: 4e – 10f = 30       – (v)
    • Subtract (v) from (iii) to eliminate terms in e: 13f = -26f
      ∴ f = -2
    • Substitute -2 for f in (ii): 2e = 5(-2) + 15
    •                                                2e = -10 + 15
    •                                                2e = 5
    • ∴ e = 21/2
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