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

 

 

 

Answer:

(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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