Question

on comparing the ratios A1/A2,b1/b2 and C1/C2 find for whether the following pair of linear equations are consistent or inconsistent. a) 4/3x+2y=3; 3x-6y=1
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Answer 1

❥︎ Concept used :-

Let us consider two linear equations

$\bf \:a_1x + b_1y + c_1=0 \: and \: a_2x + b_2y + c_2=0$

❥︎ Case : - 1

$\bf \:if \: \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2} \: then$

system of equations is consistent having infinitely many solutions.

❥︎ Case :- 2

$\bf \:if \: \dfrac{a_1}{a_2} ≠\dfrac{b_1}{b_2} \: then$

system of equations is consistent having unique solution.

❥︎ Case :- 3

$\bf \:if \: \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} ≠ \dfrac{c_1}{c_2} \: then$

system of equations is inconsistent having no solution.

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❥︎Question

• find for whether the following pair of linear equations are consistent or inconsistent. a) 4/3x+2y=3; 3x-6y=1

❥︎ Solution :-

Here,

$\bf \:a_1 \: = \dfrac{4}{3}, b_1 = 2, \: c_1 = 3$

$\bf \:a_2 \: = 3, b_2 = - 6, \: c_2 = 1$

Now,

$\bf \:\dfrac{a_1}{a_2} = \dfrac{\dfrac{4}{3} }{3} = \dfrac{4}{9}$

$\bf \:\ \dfrac{b_1}{b_2} =\dfrac{2}{ - 6} = - \dfrac{1}{3}$

$\bf\implies \:\dfrac{a_1}{a_2} ≠\dfrac{b_1}{b_2}$

$\bf\implies \:System \: of \: equations \: is \: consistent.$