### Chapter 3: Pair of Linear Equations in Two Variables

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

Question:

On comparing the ratios  $\frac{a_1}{a_2},$   $\frac{b_1}{b_2}$  and  $\frac{c_1}{c_2}$  , find out whether the following pair of linear equations are consistent, or inconsistent.

(i)  3x + 2y  = 5; 2x - 3y  = 7

(ii)  2x - 3y  = 8; 4x - 6y  = 9

(iii)  $\frac{3}{2}x + \frac{5}{3}y = 7$ ; 9x - 10y = 14

(iv)  5x - 3y  = 11; -10x + 6y  = -22

(v)  $\frac{4}{3}x + 2y = 8$; 2x + 3y  = 12

(i)  3x + 2y  = 5; 2x - 3y  = 7

Comparing equation 3x + 2y = 5 with  $a_1x + b_1y + c_1 = 0$  and 2x − 3y = 7 with  $a_2x + b_2y + c_2 = 0$.

We get  $a_1 = 3 , b_1 = 2 , c_1 = -5$  and  $a_2 = 2, b_2 = -3 , c_2 = -7$

$\frac{a_1}{a_2} = \frac{3}{2}$    and   $\frac{b_1}{b_2} = \frac{2}{-3}$   ,  So  $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$     which means equations have a unique solution.

Hence they are consistent.

(ii)  2x - 3y  = 8; 4x - 6y  = 9

Comparing equation 2x − 3y = 8 with  $a_1x + b_1y + c_1 = 0$ and 4x − 6y = 9 with $a_2x + b_2y + c_2 = 0$ .

We get  $a_1 = 2 , b_1 = -3 , c_1 = -8$  and $a_2 = 4, b_2 = -6 , c_2 = -9$ .

We have  $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$    because of   $\frac{2}{4} = \frac{-3}{-6} \neq \frac{-8}{-9} \Rightarrow \frac{1}{2} = \frac{1}{2} \neq \frac{-8}{-9}$

Therefore, equations have no solution because they are parallel.

Hence, they are inconsistent.

(iii)  $\frac{3}{2}x + \frac{5}{3}y = 7$ ; 9x - 10y = 14

Comparing equation  $\frac{3}{2}x + \frac{5}{3}y = 7$  with  $a_1x + b_1y + c_1 = 0$ and 9x − 10y = 14 with $a_2x + b_2y + c_2 = 0$.

We get  $a_1 = \frac{3}{2} , b_1 = \frac{5}{3} , c_1 = -7$  and $a_2 = 9, b_2 = -10 , c_2 = -14$ .

$\frac{a_1}{a_2} = \frac{\frac{3}{2}}{9} = \frac{1}{6}$   and    $\frac{b_1}{b_2} = \frac{\frac{5}{3}}{-10} = \frac{-1}{6}$  , So   $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$   which means equations have a unique solution.

Hence, they are consistent.

(iv)  5x - 3y  = 11; -10x + 6y  = -22

Comparing equation 5x − 3y = 11 with $a_1x + b_1y + c_1 = 0$  and −10x + 6y = −22 with $a_2x + b_2y + c_2 = 0$ .

We get  $a_1 = 5 , b_1 = -3 , c_1 = -11$  and $a_2 = -10, b_2 = 6 , c_2 = 22$ .

$\frac{a_1}{a_2} = \frac{5}{-10} = \frac{-1}{2}$ , $\frac{b_1}{b_2} = \frac{-3}{6} = \frac{-1}{2}$ and $\frac{c_1}{c_2} = \frac{-11}{22} = \frac{-1}{2}$

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$   . Therefore, the lines have infinitely many solutions.

Hence, they are consistent.

(v)  $\frac{4}{3}x + 2y = 8$; 2x + 3y  = 12

Comparing equation $\frac{4}{3}x + 2y = 8$ with  $a_1x + b_1y + c_1 = 0$ and  2x + 3y  = 12 with  $a_2x + b_2y + c_2 = 0$.

We get  $a_1 = \frac{4}{3} , b_1 = 2 , c_1 = -8$  and  $a_2 = 2, b_2 = 3 , c_2 = -12$  .

$\frac{a_1}{a_2} = \frac{\frac{4}{3}}{2} = \frac{4}{6} = \frac{2}{3}$ , $\frac{b_1}{b_2} = \frac{2}{3}$  and  $\frac{c_1}{c_2} = \frac{-8}{-12} = \frac{2}{3}$

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.

Therefore, the lines have infinitely many solutions.

Hence, they are consistent.

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