Question

4.One equation of a pair of dependent linear equations is 2x + 5y = 3. The second equation will be *

Answer 1

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FORMULA TO BE IMPLEMENTED

A pair of linear equations of the form

$\sf{a_{1} x+b_1y=c_1 \: \: \: and \: \: \: a_2x+b_2y=c_2}$

are said to be pair of dependent linear equations if

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

GIVEN

One equation of a pair of dependent linear equations is 2x + 5y = 3

TO DETERMINE

The second equation

CALCULATION

The given equation is

$\sf{2x + 5y = 3} \: \: \: \: ......(1)$

$\sf{Comparing \: with \: \: a_{1} x+b_1y=c_1 \: \: we \: get \: \: \: a_1 = 2 \:, \: b_1=5 \: , \: c_1 = 3}$

$\sf{Let \: the \: second \: equation \: be \: \: a_2x+b_2y=c_2 \: \: ..(2)}$

Since the two lines are pair of dependent linear equations

$\therefore \: \: \displaystyle \sf{ \frac{2}{a_2} = \: \frac{5}{b_2} \: = \frac{3}{c_2} }$

$\implies \: \: \displaystyle \sf{ \frac{a_2}{2} = \: \frac{b_2}{5} \: = \frac{c_2}{3} \: \: = k \: \: (say) }$

Where k is nonzero integer

$\implies \: \: \displaystyle \sf{ {a_2} = {2k} \: , \: {b_2} = {5k} \:, \: {c_2} = {3k} \: \: }$

From Equation (2) we get

$\sf{2kx + 5ky = 3k}$

Which is the general form of the second equation of the pair of dependent linear equations with 2x + 5y = 3

For example 6x + 15y = 9 is one of the second equation

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