Question

Solve the equations.
6x + 3y = 6xy and 2x + 4y = 5xy​

Answer 1

Answer:

6x+3y-6xy=0,2x+4y-5xy=0

Answer 2

SOLUTION:-

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

•The given pair of linear equation is

$\bf 6x + 3y = 6xy$

$\bf \implies\dfrac{6x}{xy} + \dfrac{3y}{xy} = \dfrac{6xy}{xy}$[Dividing throughout by xy]

$\bf \implies\dfrac{6}{y} + \dfrac{3}{x} = 6$.....(1)

⠀⠀⠀⠀⠀⠀⠀⠀⠀

And, $\bf 6x + 3y = 6xy$

$\bf \implies\dfrac{2x}{xy} + \dfrac{4y}{xy} = \dfrac{5xy}{xy}$[Dividing throughout by xy]

$\bf \implies\dfrac{2}{y} + \dfrac{4}{x} = 5$.......(2)

Putting

$\bf \dfrac{1}{x}=u$........(3)

, $\bf \dfrac{1}{y}=v$.........(4)

⠀⠀⠀⠀⠀⠀⠀⠀⠀

Then equation (1) and (2) can be rewritten as

⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\bf 6v+3u=6$........(5)

$\bf 2v+4u=5$........(6)

•Multiplying the equation (6) by 3,we get

⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\bf 6v+12u=15$.........(7)

⠀⠀⠀⠀⠀⠀⠀⠀⠀

•Subtracting equation (5) from equation (7),we get

⠀⠀⠀⠀$\bf 9u=9$

$\bf\implies u=\dfrac{9}{9}=1$

$\bf\implies \dfrac{1}{x}=1$

$\bf \implies x=1$

•Substituting the value of u in equation (5),we get

⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\bf \implies 6u+3\times 1=6$

⠀⠀⠀⠀⠀⠀⠀⠀⠀

$\bf\implies v=\dfrac{3}{6}=\dfrac{1}{2}$

$\bf\implies \dfrac{1}{y}=\dfrac{1}{2}$

$\bf \implies y=2$

⠀⠀⠀⠀⠀⠀⠀⠀⠀

Hence ,the solution of the given pair of equation are x=1,y=2

VERIFICATION

⠀⠀⠀⠀⠀⠀⠀⠀⠀

Substituting x=1,y=2 in we find that both equation (1) and (2) are satisfied as shown below

$\bf\implies \dfrac{6}{y}+\dfrac{3}{x}= \dfrac{6}{2}+\dfrac{3}{1}=3+3=6$

$\bf\implies \dfrac{2}{y}+\dfrac{4}{x}= \dfrac{2}{2}+\dfrac{4}{1}=1+4=5$

Hence,the solution is correct