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Elimination by Comparison Method of Independent Linear Equations Simultaneously

Simultaneous System of Equations

If there is only one solution for each unknown that satisfies both equations, the two equations of the system are called Independent Simultaneous Equations in two variables.

Steps for Solving Independent Simultaneous Equations

Simultaneous equations involving two unknown quantities are solved by using following steps:

Step-I: Eliminate one of the unknowns
Step-II: Solve for the other unknown
Step-III: Find the value of the unknown previously eliminated

Method of Solving Independent Linear  Equations Simultaneously

There are following common methods of solving independent linear equations simultaneously.

  1. Elimination by addition or subtraction
  2. Elimination by substitution
  3. Elimination by comparison
  4. Use of graphs
  5. Use of matrix algebra

Elimination by Comparison Method of Independent Linear Equations Simultaneously

Elimination by Comparison

These is another method of solving systems of equations called the Comparison method. Like the Elimination By Addition or Subtraction Method and Substitution Method, it produces exact solutions and can used on any system of two linear equations involving two variables. To solve a system of two equations in two variables by Comparison Method following are the steps.

  1. Write each equation with “x or y” as the subject. We will call this the isolated-subject equation, an equation where one variable is isolated.
  2. We number the two given equations as (1) and (2).
  3. We find the value of the one of the two variables in terms of the other variable from the two given equations and number the two newly formed equations as (3) and (4).
  4. We equate the R.H.S. of the equations (3) and (4), since the L.H.S. are the same, and solve to get the numerical value of the other variable.
  5. We put this numerical value of the other variable in either equation (3) or equation (4) and solve to get the numerical value of the first variable.
  6. Check the solution by substituting for x and y in both original equations.

We consider the following example to illustrate the use of the Comparison Method.

Question:-  3x + 2y = 54 and 2x – 3y = 10

Solution:-

Given Set of equations:

3x + 2y = 54             ………(i)
2x – 3y = 10             ………(ii)

Step-I:  Here, we find the value of “x” or isolated the “x” in the first and second equation from the two given equations and number the two newly formed equations as (iii) and (iv).

      x =                  ………(iii)
x =                  ………(iv)

Step-II: Equating equation (iii) in x with equation (iv) in x

        = 
2(54 – 2y) = 3(10 + 3y)
108 – 4y = 30 + 9y
108 – 30 = 9y + 4y
78 = 13y
     y =   = 6

Step-III: Placing this value of y = 6 in equation (iii) we get x

      x =                  ………(iii)
=
=
= 
= 14

Thus the solution set of the given system of equations is

S.S = {(x, y)} = {(14 , 6)}

To Check: Substitute x = 14 and y = 6 in both of the original equations

3x + 2y = 54             ………(i)
3(14) + 2(6) = 54
42 + 12 = 54
54 = 54 (a true statement)

2x – 3y = 10             ………(ii)
2(14) – 3(6) = 10
28 – 18 = 10
10 = 10 (a true statement)

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