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Previous Next##### Pair of linear equations in two variables CBSE NCERT Solutions

Question:

**Which of the following pairs of linear equations has a unique solution, no solution, or infinitely many solutions? In case there is a unique solution, find it by using cross multiplication method.**

**(i) x – 3y – 3 = 0**

** 3x – 9y – 2 = 0**

**(ii) 2x + y = 5**

** 3x + 2y = 8**

**(iii) 3x − 5y = 20**

** 6x − 10y = 40**

**(iv) x − 3y – 7 = 0**

** 3x − 3y – 15 = 0**

Answer:

**(i) ****x ****− 3y – 3 = 0**

** 3x − 9y – 2 = 0**

**We compare equation x − 3y – 3 = 0 with a_{1}x +b_{1}y + c_{1} = 0 and 3x − 9y – 2 = 0 with a_{2}x +b_{2}y + c_{2} = 0.**

**We get a _{1 }= 1, b_{1}= -3, c_{1} = -3 and a_{2} = 3, b_{2 }= -9, c_{2} = -2.**

**Here ** **this means that the two lines are parallel.**

**Therefore, there is no solution for the given equations i.e. it is inconsistent.**

**(ii) ****2x + y = 5**

** 3x + 2y = 8**

**Comparing equation 2 x + y = 5 with a_{1}x + b_{1}y + c_{1} = 0 and 3x + 2y = 8 with a_{2}x + b_{2}y + c_{2} = 0.**

**We get a _{1 }= 2, b_{1}= 1, c_{1} = -5 and a_{2} = 3, b_{2 }= 2, c_{2} = -8**

**Here ** **this means that there is a unique solution for the given equations.**

**⇒** *x ***= 2 and y = 1**

**(iii) ****3x − 5y = 20**

** 6x − 10y = 40**

**Comparing equation 3 x − 5y = 20 with a_{1}x + b_{1}y + c_{1} = 0 and 6x − 10y = 40 with a_{2}x + b_{2}y + c_{2} = 0.**

**We get a _{1 }= 3, b_{1}= -5, c_{1} = -20 and a_{2} = 6, b_{2 }= -10, c_{2} = -40**

**Here,**

**It means lines coincide with each other.**

**Hence, there are infinitely many solutions.**

**(iv) ****x ****− 3y – 7 = 0**

** 3x − 3y – 15 = 0**

**Comparing equation x − 3y – 7 = 0 with a_{1}x + b_{1}y + c_{1} = 0 and 3x − 3y – 15 = 0 with a_{2}x + b_{2}y + c_{2} = 0.**

**We get a _{1 }= 1, b_{1}= -3, c_{1} = -7 and a_{2} = 3, b_{2 }= -3, c_{2} = - 15**

**Here ** **this means that we have a unique solution for these equations.**

**⇒** *x ***= 4 and y = –1**

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