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Question:

**Solve the following pair of linear equations by the substitution method.**

**(i) x + y = 14****; x – y = 4**

**(ii) s – t = 3; **

**(iii) 3x – y = 3****; 9x − 3y = 9**

**(iv) 0.2x + 0.3y = 1.3****; 0.4x + 0.5y = 2.3**

**(v) **** ****; **

**(vi) ; **

Answer:

**(i)**** x + y = 14 …(1)**

** x ****– y = 4 … (2)**

**From equation (2), x – y = 4 ****⇒**** x = 4 + y **

**Putting x = 4 + y in equation (1), we get**

** 4 + y + y = 14**

**⇒**** 2y = 10 ****⇒** **y ****= 5**

**Putting value of y in equation (1), we get**

** x ****+ 5 = 14**

**⇒** **x ****= 14 – 5 = 9**

**Therefore, x = 9 and y = 5**

**(ii)**** s – t = 3 … (1)**

** **** …(2)**

**Using equation (1), we can say that s = 3 + t**

**Putting s = 3 + t in equation (2), we get**

**⇒ ****6 + 2t + 3t = 36**

**⇒**** 5t + 6 = 36**

**⇒**** 5t = 30 ****⇒** **t ****= 6**

**Putting value of t = 6 in equation (1), we get**

**s ****– 6 = 3 ****⇒** **s ****= 3 + 6 = 9**

**Therefore, t = 6 and s = 9**

**(iii)**** 3x – y = 3 … (1)**

** 9x − 3y = 9 … (2)**

**Comparing equation 3x – y = 3 with and equation 9x − 3y = 9 with .**

**We get and **

**Here,**

**Therefore, we have infinite many solutions for x and y**

**(iv)**** 0.2x + 0.3y = 1.3 … (1)**

** 0.4x + 0.5y = 2.3 … (2)**

**Using equation (1), we can say that**

**0.2x = 1.3 − 0.3y**

**⇒** **x ****= **

**Putting this in equation (2), we get**

**⇒**** 2.6 ****−**** 0.6y + 0.5y = 2.3**

**⇒** **−****0.1y = −0.3 ****⇒** **y ****= 3**

**Putting value of y in (1), we get**

**0.2x + 0.3 (3) = 1.3**

**⇒**** 0.2x + 0.9 = 1.3**

**⇒**** 0.2x = 0.4 ****⇒** **x ****= 2**

**Therefore, x = 2 and y = 3**

**(v) **** **** ….(1)**

** ** ** ….(2)**

**Using equation (1), we can say that**

**Putting this in equation (2), we get**

** **

** **

**⇒** **y ****= 0**

**Putting the value of y in (1), we get x = 0**

**Therefore, x = 0 and y = 0**

**(vi) … (1)**

** … (2)**

**Using equation (2), we can say that**

**Putting this in equation (1), we get**

** **

** **

**⇒** *y ***= 3**

**Putting the value of y in equation (2), we get**

** **

** **

**⇒**** x = 2**

**Therefore, x = 2 and y = 3**

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