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Find the root of 2x-3/x-2+2x-7/x-4=16/3 |
| Answer» {tex}Question:\\space\\large{ {2x-3\\over x-2} +{2x-7\\over x-4} = {16\\over 3}, find \\space x?}{/tex}{tex}\\large {Solution:\\\\}{/tex}\t{tex}\\large {\\underline {Step\\space 1:\\space Fraction\\space solving}}{/tex}{tex}\\large\\implies {{\\big((2x-3)(x-4)\\big)} +{\\big((2x-7)(x-2)\\big)}\\over {(x-2)(x-4)}} = {16\\over 3}{/tex}{tex}\\large\\implies {{\\big((2x^2-8x-3x+12)\\big)} +{\\big((2x^2-4x-7x+14)\\big)}\\over {(x^2-4x-2x+8)}} = {16\\over 3}{/tex}{tex}\\large\\implies {{\\big(2x^2-11x+12+2x^2-11x+14\\big)}\\over {(x^2-6x+8)}} = {16\\over 3}{/tex}{tex}\\large\\implies {{\\big(4x^2-22x+26\\big)}\\over {(x^2-6x+8)}} = {16\\over 3}{/tex}{tex}\\large{ \\underline {Step\\space 2:\\space Cross-multiplication}\\\\\\implies {{3\\times \\big (4x^2-22x+26\\big)} } = {16\\times {(x^2-6x+8)}}}{/tex}\xa0{tex}\\large{ \\underline {Step\\space 3:\\space Simplification:}\\\\\\implies {{\\big (12x^2-66x+78\\big)} } = {{(16x^2-96x+128)}}}{/tex}\xa0{tex}\\large {\\underline {Step\\space 4:\\space Equation-solving:}\\\\\\implies 16x^2-96x+128-12x^2+66x-78=0}{/tex}{tex}\\large {\\underline {Step\\space 5:\\space Arranging\\space like-terms:}\\\\\\implies 16x^2-12x^2-96x+66x+128-78=0}{/tex}{tex}\\large{\\\\\\implies 4x^2-30x+50=0}{/tex}\xa0{tex}\\large {\\underline {Step\\space 6:\\space Taking\\space out\\space common-terms:}\\\\\\implies 2\\times (2x^2-15x+25)=0}{/tex}{tex}\\large {\\\\\\implies 2x^2-15x+25=0}{/tex}{tex}\\large {\\underline {Step\\space 7:\\space Middle-term\\space splitting\\space :}\\\\\\implies 2x^2-10x-5x+25=0}{/tex}{tex}\\large{ \\underline {Step\\space 8:\\space Grouping\\space :}\\\\\\implies 2x(x-5)-5(x-5)=0}{/tex}{tex} \\\\\\implies \\large(2x-5)(x-5)=0{/tex}{tex} \\\\\\implies\\huge \\therefore \\boxed{x=5\\space or \\space {5\\over2}}{/tex}\t\xa0 | |