**Substitution method is an algebraic method of solving the system of linear equations. Using this method, we can find the exact solution for the equations.**

**Let us explain this method using few examples.**

**Example 1:**

**Solve the system of linear equations using substitution method.**

*y*= 2*x*

*x*+ 2*y*= 5**Solution:**

**Step 1:**

**Given equations are**

*y*= 2*x*and*x*+ 2*y*= 5.**The value of**

*y*is directly given in the first equation. So, substitute the value of*y*= 2*x*in the second equation*x*+ 2*y*= 5 and simplify.**So,**

*x*+ 2*y*= 5 becomes*x*+ 2(2*x*) = 5.

*x*+ 4*x*= 5**Add 4**

*x*and*x*.**5**

*x*= 5**Divide by 5 on both the sides to isolate**

*x*.

*x*= 1**Step 2:**

**Substitute the value**

*x*= 1 to find the value of*y*.**Replace the value of**

*x*by 1 in the equation*y*= 2*x*.

*y*= 2 (1)

*y*= 2**Step 3:**

**So, the solution of the given system of equations is (1, 2).**

**Example 2:**

**Using substitution method, solve the given linear equations.**

*x*–*y*= 3**3**

*x*+ 2*y*= 9**Solution:**

**Step 1:**

**Given equations are**

*x*-*y*= 3 and 3*x*+ 2*y*= 9.**We need to find the value of variables by substituting the value of one variable in the other equation.**

**Let us find the value of**

*x*from the first equation.**To find the value of**

*x*, solve for*x*.**Add**

*y*on both the sides of the equation*x*-*y*= 3 to find the value of*x*and then simplify.

*x*-*y*+*y*= 3 +*y*

*x*=*y*+ 3**Step 2:**

**Replace the value of**

*x*by*y*+ 3 in the second equation 3*x*+ 2*y*= 9 and simplify it.**3(**

*y*+ 3) + 2*y*= 9**3y + 9 + 2**

*y*= 9**Combine the like terms.**

**(3**

*y*+ 2*y*) + 9 = 9**5**

*y*+ 9 = 9**Subtract 9 from both the sides.**

**5**

*y*+ 9 - 9 = 9 - 9**5**

*y*= 0**Divide by 5 on both the sides.**

*y*= 0**Step 3:**

**Now, substitute the value of**

*y*= 0 in any of the equation whichever is easy to solve.**So, substitute the value of**

*y*in the first equation*x*-*y*= 3.

*x*- 0 = 3

*x*= 3**Step 4:**

**So, the solution is (3, 0).**

**Example 3:**

**Use substitution method to solve the given system of linear equations.**

**2**

*x*+*y*= 4**4**

*x*+ 2*y*= 8**Solution:**

**Step 1:**

**Given equations are 2**

*x*+*y*= 4 and 4*x*+ 2*y*= 8.**We need to find the value of variables by substituting the value of one variable in the other equation.**

**Let us find the value of**

*y*from the first equation 2*x*+*y*= 4.**Subtract 2**

*x*from both the sides,**2**

*x*- 2*x*+*y*= 4 - 2*x*

*y*= 4 - 2*x***Step 2:**

**Now, substitute the value of**

*y*= 4 - 2*x*in the second equation 4*x*+ 2*y*= 8.**4**

*x*+ 2(4 - 2*x*) = 8**4**

*x*+ 8 - 4*x*= 8**8 = 8**

**Step 3:**

**We end up with a statement 8 = 8. This is a true statement which means that the given system of equations has infinitely many solutions.**

**Important note:**

**While solving the system of equations, we may arrive at some statements like 5 = 5, -5 = -5, and 3 = 4.**

**If we arrive at a true statement (E.g. 4 = 4), then the system has infinitely many solutions and the graph of both the equations are the same.**

**If we arrive at a false statement (E.g. 4 = 3), then the system has no solution and the lines are parallel.**

**Practice questions:**

**Use substitution method to solve the given system of equations.**

**a) -2x + 9y = -9; 9x – 2y = 2**

**b) x + 2y = -3; -2x + 4y = 10**

**c) x + y = 3; x + y = 1**

**d) x + 5y = 8; 2x – 3y = -10**

**e) 7x – y = 10; 9x + 2y = 3**

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