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Solve the following pairs of equations by reducing them to a pair of linear equations: (vi) 6x + 3y = 6xy 2x + 4 y =- 5 xy

Q1.    Solve the following pairs of equations by reducing them to a pair of linear equations:

                (vi)    \\6x + 3y = 6xy\\ 2x + 4y = 5 xy

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Given Equations,

\\6x + 3y = 6xy\\\Rightarrow \frac{6x}{xy}+\frac{3y}{xy}=6\\\\\Rightarrow \frac{6}{y}+\frac{3}{x}=6\\and\\\ 2x + 4y = 5 xy\\\Rightarrow \frac{2x}{xy}+\frac{4y}{xy}=5\\\Rightarrow \frac{2}{y}+\frac{4}{x}=5

Let, 

\frac{1}{x}=p\:and\:\frac{1}{y}=q

Now, our equation becomes

6q+3p=6........(1)

And

2q+4p=5..........(2)

By Cross Multiplication method,

\frac{q}{b_1c_2-b_2c_1}=\frac{p}{c_1a_2-c_2a_1}=\frac{1}{a_1b_2-a_2b_1}

\frac{q}{(3)(-5)-(-6)(4)}=\frac{p}{(6)(2)-(6)(-5)}=\frac{1}{(6)(4)-(3)(2)}

\frac{q}{-15+24}=\frac{p}{-12+30}=\frac{1}{24 -6}

\frac{q}{9}=\frac{p}{18}=\frac{1}{18}

q=\frac{1}{2}\:and\:p=1

And Hence,

x=1\:and\:y=2.

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