Question

$check whether (3/2)x + (5/3y)=7,9x-10y=12 is consistent or inconsistent.solve them graphically$
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Answer 1

FORMULA TO BE IMPLEMENTED :

For A pair of Straight Lines

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

CONDITION FOR CONSISTENT

1. Coincident lines :

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

2. Intersecting lines :

$\displaystyle \: \: \frac{a_1}{a_2} \ne \frac{b_1}{b_2}$

3. Parallel lines ( Inconsistent) :

$\displaystyle \: \: \frac{a_1}{a_2} = \frac{b_1}{b_2}</strong><strong> </strong><strong> </strong><strong>\</strong><strong>ne</strong><strong> </strong><strong>\frac{c_1}{c_2}$

CALCULATION

Given pair of linear equations

$\displaystyle \: \frac{3x}{2} + \frac{5y}{3} = 7$

9x + 10y = 42 - - - - - - - (1)

9x - 10y = 12 - - - - - - - - (2)

Comparing with

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

We get

$\displaystyle \: a_1 = 9 \: , \: b_1 = 10</p><p> \: , c_1= - 42\: and \: \: a_2 = 9 \: , \: b_2 = - 10\: , \: \: c_2= - 12$

Now

$\displaystyle \: \: \frac{a_1}{a_2} = \frac{9}{9} =1$

$\displaystyle \: \: \frac{b_1}{b_2} = \frac{10}{-10}=-1$

$\displaystyle \: \: \frac{c_1}{c_2} = \frac{-42}{-12}= \frac{7}{2 }$

So

$\displaystyle \: \: \frac{a_1}{a_2} \ne \frac{b_1}{b_2}$

From above we can conclude that the given lines are intersecting

Hence the given lines are consistent

Graph : For graph refer to the attachment