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Introduction to Linear Simultaneous Equations

Simultaneous Equations

A set of linear equations in two variables can be solved simultaneously and hence are called simultaneous equations.
ax + by = k
cx + dy = l
    Simultaneous equations can be solved by
  1. Substitution Method
  2. Elimination Method
In both the above methods
  1. Value of one variable is found
  2. Value is substituted to obtain the value of other variable

Substitution Method


In substitution method we express one variable in terms of the other and substitute it to get value of one variable. Then substitute it in any equation to get value of other variable.

  • Step 1:

    • Express any one equation in terms of one variable.
      2x−y = 3 ⇒ y = 2x − 3
  • Step 2:

    • Substitute it in next equation.
      4x + y = −6
      ⇒ 4x + 2x − 3 = −6
      ⇒ 6x = −3
      ⇒ x = −(1/2) = −.5
  • Step 3:

    • Substitute value in the first equation.
      y = 2x − 3 = 2(−.5) − 3 = −4
      So solution is x = −.5 and y = −4

Elimination Method

The elimination method is based on the fact that if one variable is removed from the equation then the reduced equation contains only one variable. After we can find the solution. Substitute one solution in one equation to get the other solution.

  • Step 1:
    • Eliminate one variable by making coefficient of that variable equal in two equations.
      2x − y = 3
      4x + y = −6
      Add the two equations
      6x = −3
  • Step 2:
    •  x = −(1/2) = −.5
  • Step 3:
    • Substitute in anyone
      2(−.5) − y = 3
      y = − 4

Further Reading

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