Question

show that the linear equation 4x+7y-2=2 and 8x+14 y=7 is inconsistent​

Answer 1

SOLUTION

TO PROVE

The linear equation 4x+7y-2=2 and 8x+14 y=7 is inconsistent

CONCEPT TO BE IMPLEMENTED

A pair of Straight Lines

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

is said inconsistent if

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

EVALUATION

Here the given equations are

4x+7y-2=2 and 8x+14 y=7

Which can be rewritten as

4x + 7y - 4 = 0 - - - - (1)

8x + 14y - 7 = 0 - - - -(2)

Comparing with the lines

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

We have

$\displaystyle \sf{ a_1 = 4 \: , \: b_1 = 7 , c_1= - 4 \: and \: \: a_2 = \: 8 , \: b_2 = 17\: , \: \: c_2= - 7}$

Now

$\displaystyle \sf{ \: \frac{a_1}{a_2} = \frac{4}{8} \ = \: \frac{1}{2}}$

$\displaystyle \sf{ \: \frac{b_1}{b_2} = \frac{7}{14} = \frac{1}{2} }$

$\displaystyle \sf{ \frac{c_1}{c_2} = \frac{ - 4}{ - 7} = \frac{4}{7} }$

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

So the given system of equations are inconsistent

Hence proved

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