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  • (i)Are the following pair of linear equations consistent Justify your answer. –3x– 4y = 12 ; 4y + 3x = 12

Are the following pair of linear equations consistent? Justify your answer.

(i) –3x– 4y = 12 ; 4y + 3x = 12

(ii) \frac{3}{5}x-y= \frac{1}{2};\frac{1}{5}x-3y= \frac{1}{6} ;
(iii) 2ax + by = a; 4ax + 2by = 2a ; a, b ≠ 0
(iv) x + 3y = 11 ; 4x + 12y = 22

 

 

Answers (1)

Solution: (i)
Equation are : –3x – 4y = 12
4y + 3x = 12
In equation
–3x – 4y – 12 = 0
a1 = –3; b1 = –4 ; c1 = –12
In equation
4y + 3x – 12 = 0
a2 = 3 ; b2 = 4; c2 = –12
\frac{a_{1}}{a_{2}}= \frac{-3}{3}= -1;\frac{b_{1}}{b_{2}}= \frac{-4}{4}= -1,\frac{c_{1}}{c_{2}}\Rightarrow \frac{-12}{-12}= \frac{1}{1}

For consistency either \frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}or \frac{a_{1}}{a_{2}}= \frac{b_{1}}{b_{2}}= \frac{c_{1}}{c_{2}} but here \frac{a_{1}}{a_{2}}= \frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}

Hence the pair of linear equations is not consistent.
Solution: (ii)
Equation are : \frac{3}{5} x-y=\frac{1}{2}

\frac{1}{5} x-3 y=\frac{1}{6}
In equation 

\frac{3}{5} x-y=\frac{1}{2}
a1= \frac{3}{5}; b1 = –1 ; c1 = -\frac{1}{2}
In equation

\frac{1}{5} x-3 y=\frac{1}{6}

a2 = 1/5 ; b2 = – 3; c2 = –1/6

\frac{a_{1}}{a_{2}}= \frac{3}{5}\times \frac{5}{1}= 3;\frac{b_{1}}{b_{2}}= \frac{-1}{-3}= \frac{1}{3};\frac{c_{1}}{c_{2}}= \frac{-1}{-2}\times \frac{6}{1}= 3

For consistency either \frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}or\frac{a_{1}}{a_{2}}= \frac{b_{1}}{b_{2}}= \frac{c_{1}}{c_{2}} also here \frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}

Hence given equations are consistent.  
Solution: (iii)
Equation are:   2ax + by = a
4ax + 2by = 2a
In equation
2ax + by – a = 0
a1 = 2a; b1 = b; c1 = –a
In equation
4ax + 2by – 2a = 0
a2 = 4a ; b2 = 2b; c2 = –2a

\frac{a_{1}}{a_{2}}= \frac{2a}{4a}= \frac{1}{2};\frac{b_{1}}{b_{2}}= \frac{b}{2b}= \frac{1}{2};\frac{c_{1}}{c_{2}}= \frac{-a}{-2a}= \frac{1}{2}
For consistency either \frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}or \frac{a_{1}}{a_{2}}= \frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}} 

Also, here \frac{a_{1}}{a_{2}}= \frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}

Hence given equations are consistent
Solution: (iv)

Given, equations are   x + 3 y = 11
4x + 12y =22
In equation
x + 3y = 11
a1 = 1; b1 = 3; c1 = –11
In equation
4x + 12y = 22
a2 = 4 ; b2 = 12; c2 = –22
\frac{a_{1}}{a_{2}}= \frac{1}{4} ;\frac{b_{1}}{b_{2}}=\frac{3}{12}= \frac{1}{4};\frac{c_{1}}{c_{2}} = \frac{-11}{-22}= \frac{1}{2}
For consistency either \frac{a_{1}}{a_{2}}\neq \frac{b_{1}}{b_{2}}or \frac{a_{1}}{a_{2}}= \frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}} but here \frac{a_{1}}{a_{2}}= \frac{b_{1}}{b_{2}}\neq \frac{c_{1}}{c_{2}}

Hence given equations are not consistent.

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