Question

find whether the lines representing the following pair of linear equations consistent or inconsistent, 3x+2y=3, x-2y=6​

Answer 1

       $\underline{\bold{Solution:}}$

       $\text{The given pair of linear equations are:}$

       $3x + 2y = 3 \text{ ------ (i)}$

       $x - 2y = 6 \text{ ------ (ii)}$

       $\text{First, we must check the values of }\dfrac{a_1}{a_2}, \dfrac{b_1}{b_2}, \dfrac{c_1}{c_2}$

       $\text{In our case, }a_1 = 3, b_1 = 2, c_1 = 3 \text{ and }a_2 = 1, b_2 = -2, c_2 = 6$

$\implies \dfrac{a_1}{a_2} = \dfrac{3}{1}$

$\implies \dfrac{b_1}{b_2} = \dfrac{2}{-2} = -\dfrac{1}{1}$

$\implies \dfrac{c_1}{c_2} = \dfrac36 = \dfrac12$

       $\text{We can see that }\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}$

$\implies \boxed{\text{(i) and (ii) are consistent as they have a unique solution}}$