Question

4. On comparing the ratios ai/az, b1/b2, and ci/cz, 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​

Answer 1

Basic Concept Used :-

Let us consider pair of lines as

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

1. System of equation is consistent having unique solutuon

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

2. System of equations is consistent having infinitely many solutions iff

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

3. System of equations is inconsistent having no solution iff

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

Let's solve the problem now!!!

Answer (i)

Given pair of lines are

• 3x + 2y = 5

• 2x - 3y = 7

$\red{ \sf \: On \: comaring \: with \: a_1x + b_1y + c_1 = 0 and a_2x + b_2y + c_2 = 0}$

We get,

$\sf \: a_1 = 3 \\ \sf \: b_1 = 2 \\ \sf \: c_1 = 5 \\ \sf \: a_2 = 2 \\ \sf b_2 = - 3 \\ \sf \: c_2 = 7$

Now,

$\bf \:\dfrac{a_1}{a_2} = \dfrac{3}{2}$

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

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

Hence, System of equations is consistent having unique solution.

Answer :- (ii)

Given equations of lines are

• 2x - 3y = 8

• 4x - 6y = 9

$\red{ \sf \: On \: comaring \: with \: a_1x + b_1y + c_1 = 0 and a_2x + b_2y + c_2 = 0}$

we get

$\sf \: a_1 = 2 \\ \sf \: b_1 = - 3 \\ \sf \: c_1 = 8 \\ \sf \: a_2 = 4 \\ \sf b_2 = - 6 \\ \sf \: c_2 = 9$

Now,

$\bf \:\dfrac{a_1}{a_2} = \dfrac{ 2}{4} = \dfrac{1}{2}$

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

$\bf \:\dfrac{c_1}{c_2} = \dfrac{8}{9}$

$\bf\implies \:\:\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \: \ne \: \dfrac{c_1}{c_2}$

Hence, System is inconsistent having no solutions.