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| 1. |
Solve the following equation:4/x +5y=7 and 3/x + +4 y=5 |
| Answer» The given system of equations is{tex}\\frac{4}{x} + 5y = 7{/tex} ....(1){tex}\\frac{3}{x} + 4y = 5{/tex} .....(2)Put {tex}\\frac{1}{x} = X{/tex} ....(3)Then equations (1) and (2) can be rewritten as4X + 5y = 7 ....(4)3X + 4y = 5 .....(5){tex}\\Rightarrow{/tex} 4X + 5y - 7 = 0 ....(6)3X + 4y - 5 = 0 .....(7)Then,{tex}\\frac{X}{{(5)( - 5) - ( 4)( - 7)}} = \\frac{y}{{( - 7)(3) - ( - 5)(4)}}{/tex}{tex}= \\frac{1}{{(4)(4) - (3)(5)}}{/tex}{tex}\\Rightarrow \\;\\frac{X}{{ - 25 + 28}} = \\frac{y}{{ - 21 + 20}} = \\frac{1}{{16 - 15}}{/tex}{tex}\\Rightarrow \\;\\frac{X}{3} = \\frac{y}{-1} = \\frac{1}{1}{/tex}{tex}\\Rightarrow{/tex} X = 3 and y = -1{tex} \\Rightarrow \\;\\frac{1}{x} = 3{/tex} and y = -1 ....using (3){tex}\\Rightarrow \\;x = \\frac{1}{3}{/tex} and y = -1Hence, the solution of the given system of equations is{tex}x = \\frac{1}{3}{/tex}, y = -1Verification : Substituting {tex}x = \\frac{1}{3}{/tex}, y = -1,We find that both the equations (1) and (2) are satisfied as shown below{tex}\\frac{4}{x} + 5y = \\frac{4}{{\\left( {\\frac{1}{3}} \\right)}} + 5( - 1) = 12 - 5 = 7{/tex}{tex}\\frac{3}{x} + 4y = \\frac{3}{{\\left( {\\frac{1}{3}} \\right)}} + 4( - 1) = 9 - 4 = 5{/tex}Hence, the solution of the given system of equations is {tex}x = \\frac{1}{3}{/tex}, y = -1 | |