Elimination by Substitution 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 Substitution

These is another method of solving systems of equations called the substitution method. Like the Elimination By Addition or Subtraction Method and Comparison 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 Substitution, following are the steps.

1. Solve one of the equations for one of the variables in terms of the other.
2. Substitute this expression into the other equation to give one equation in one unknown.
3. Solve this linear equation for the unknown.
4. Substitute this solution into the equation in step (1) or into one of the original equations to solve for other variable.
5. Check the solution by substituting for x and y in both original equations.

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

Question:-  2x + y = 5 and 3x – 2y = 11

Solution:-

Given Set of equations:

2x + y = 5                 ………(i)
3x – 2y = 11             ………(ii)

Step-I: We can eliminate “y” from equation (i) by placing value of y taken from equation (ii)

3x – 2y = 11             ………(ii)
= y            ………(ii)

Step-II: Substituting y in equation (i)

2x + y = 5                 ………(i)
2x +   = 5
= 5
=5
7x – 11 = 5 x 2
7x = 10 + 11
7x = 21
 x =  = 3

Step-III: We can get value of y by substituting x = 3 in equation (ii)

3x – 2y = 11             ………(ii)
3(3) – 2y = 11
9 – 2y = 11
– 2y = 11 – 9
y =
y = -1

Thus the solution set of the given system of equations is

S.S = {(x, y)} = {(3, -1)}

To Check: Substitute x = 3 and y = -1 in both of the original equations

2x + y = 5                 ………(i)
2(3) + 1(-1) = 5
6 – 1 = 5
5 = 5 (a true statement)

3x – 2y = 11               ………(ii)
3(3) – 2(-1) = 11
9 + 2= 11
11 = 11 (a true statement)