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Solve the equation 6x+2x=6xy 4x+3y=5xy

Answer» The given pair of equation is6x + 3y = 6xy{tex}\\Rightarrow \\quad \\frac { 6 x } { x y } + \\frac { 3 y } { x y } = \\frac { 6 x y } { x y }{/tex}..............Dividing throughout by xy{tex}\\Rightarrow \\quad \\frac { 6 } { y } + \\frac { 3 } { x } = 6{/tex}.........................(1)2x + 4y = 5xy{tex}\\Rightarrow \\quad \\frac { 2 x } { x y } + \\frac { 4 y } { x y } = \\frac { 5 x y } { x y }{/tex} ..................Dividing throughout by xy{tex}\\Rightarrow \\quad \\frac { 2 } { y } + \\frac { 4 } { x } = 5{/tex}..........................(2)Put\xa0{tex}\\frac { 1 } { x } = u{/tex} .....................(3)And\xa0{tex}\\frac { 1 } { y } = v{/tex}..........................(4)Then, the equation (1) and (2) can be written as:{tex}6 v + 3 u = 6{/tex}..................(5)2 v + 4 u = 5 ..................(6)Multiplying equation (6) by 3, we get6 v + 12 u = 15 ..........................(7)Subtracting equation (5) from equation(7), we get 9u = 9\xa0{tex}\\Rightarrow \\quad u = \\frac { 9 } { 9 } = 1{/tex}{tex}\\Rightarrow \\quad \\frac { 1 } { x } = 1{/tex}.......................using (3){tex}\\Rightarrow \\quad x = 1{/tex}Substituting this value of u in equation (5), we get 6v + 3 X 1 = 6{tex}\\Rightarrow \\quad 6 v + 3 = 6{/tex}{tex}\\Rightarrow 6 v = 6 - 3 = 3{/tex}{tex}\\Rightarrow \\quad v = \\frac { 3 } { 6 } = \\frac { 1 } { 2 }{/tex}{tex}\\Rightarrow \\quad \\frac { 1 } { y } = \\frac { 1 } { 2 }{/tex}......................using (4){tex}\\Rightarrow y =2{/tex}Hence, the solution of the given pair of the equation is x =1, y =2Verification: Substituting x =1, y =2We find that both the equation (1) and (2) are satisfied as shown below:{tex}\\frac { 6 } { y } + \\frac { 3 } { x } - \\frac { 6 } { 2 } + \\frac { 3 } { 1 } - 3 + 3 - 6{/tex}{tex}\\frac { 2 } { y } + \\frac { 4 } { x } - \\frac { 2 } { 2 } + \\frac { 4 } { 1 } = 1 + 4 = 5{/tex}Hence, the solution is correct.


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