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Ingrese su respuesta como números enteros o bien  fracciones.

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$x=$ [[0]]

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$y=$ [[1]]

\n

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Resolver el siguiente sistema de Ecuaciones:

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\$\\begin{eqnarray*}\\simplify{{a1}(x+{b1})/{c1}+{d1}(y+{f1})/{g1}}&=&\\var{h1}\\\\\\\\\\simplify{{a2}(x+{b2})/{c2}+{d2}(y+{f2})/{g2}}&=&\\var{h2}\\end{eqnarray*}\$

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En la primera ecuación multiplicamos todos los término por el Mínimo Común Multiplo de sus denomimadores que es  $\\simplify{{m}}$,  y para la segunda ecuación repetimos lo anterior pero con el MCM de sus respectivos denominadores que es  $\\simplify{{n}}$, nos queda:

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\\begin{align}
\\simplify{{m}}\\cdot\\simplify{{a1}(x+{b1})/{c1}+ {d/d1}{m}}\\cdot\\simplify{{d}(y+{f1})/{g1}} &= \\simplify{{m}}\\cdot\\var{h1} \\\\
\\simplify{{n}}\\cdot\\simplify{{a2}(x+{b2})/{c2}+ {d/d2}{n}}\\cdot\\simplify{{d}(y+{f2})/{g2}} &= \\simplify{{n}}\\cdot\\var{h2}
\\end{align}

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Multiplicamos y simplificamos:

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\\begin{align}
\\simplify{{m}{a1}(x+{b1})/{c1}+{m}{d1}(y+{f1})/{g1}} &= \\simplify{{m*h1}} \\\\
\\simplify{{n}{a2}(x+{b2})/{c2}+{n}{d2}(y+{f2})/{g2}} &= \\simplify{{n*h2}}
\\end{align}

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Volvemos a multiplicar, reducimos y ordenamos:

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\\begin{align}
\\simplify{{m}{a1}(x)/{c1}+{m}{d1}(y)/{g1}} &= \\simplify{{m*h1}-{m}{a1}({b1})/{c1}-{m}{d1}({f1})/{g1}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(*)\\\\
\\simplify{{n}{a2}(x)/{c2}+{n}{d2}(y)/{g2}} &= \\simplify{{n*h2}-{n}{a2}({b2})/{c2}-{n}{d2}({f2})/{g2}}
\\end{align}

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Una alternativa es multiplicar la primera ecuación por $\\simplify{{p/((m*d)/g1)}}$ y la segunda ecuación por $\\simplify{{p/((n*d)/g2)}}$ y así podemos cancelar la variable $y$. Nos queda:

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\\begin{align}
\\simplify{{p/((m*d)/g1)}{m}{a1}(x)/{c1}+{p/((m*d)/g1)}{m}{d1}(y)/{g1}} &= \\simplify{{p/((m*d)/g1)}({m*h1}-{m}{a1}({b1})/{c1}-{m}{d1}({f1})/{g1})} \\\\
\\simplify{{p/((n*d)/g2)}{n}{a2}(x)/{c2}+{p/((n*d)/g2)}{n}{d2}(y)/{g2}} &= \\simplify{{p/((n*d)/g2)}({n*h2}-{n}{a2}({b2})/{c2}-{n}{d2}({f2})/{g2})}
\\end{align}

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Sumando ambas ecuaciones:

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\\begin{align}
\\simplify{{p/((m*d)/g1)}{m}{a1}/{c1}+{p/((n*d)/g2)}{n}{a2}/{c2}}\\simplify{x} &= \\simplify{{p/((m*d)/g1)}({m*h1}-{m}{a1}({b1})/{c1}-{m}{d1}({f1})/{g1})+{p/((n*d)/g2)}({n*h2}-{n}{a2}({b2})/{c2}-{n}{d2}({f2})/{g2})}
\\end{align}

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Resolvemos para obtener el valor de $x$

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\\begin{align}
\\simplify{x} &= \\simplify{({p/((m*d)/g1)}({m*h1}-{m}{a1}({b1})/{c1}-{m}{d1}({f1})/{g1})+{p/((n*d)/g2)}({n*h2}-{n}{a2}({b2})/{c2}-{n}{d2}({f2})/{g2}))/{{{p}/(({m}*d)/g1)}{m}*{a1}/{c1}+{p/((n*d)/g2)}{n}*{a2}/{c2}}}
\\end{align}

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Finalmente reemplazamos el valor de $x$ en la ecuación $\\,(*)\\,$ y así obtenemos el valor de $y$:

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\\begin{align}
\\simplify{{m}{a1}(x)/{c1}+{m}{d1}(y)/{g1}} &= \\simplify{{m*h1}-{m}{a1}({b1})/{c1}-{m}{d1}({f1})/{g1}}
\\end{align}

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\\begin{align}
\\simplify{(({(m*a1)/c1}))({p/((m*d)/g1)}({m*h1}-{m}{a1}({b1})/{c1}-{m}{d1}({f1})/{g1})+{p/((n*d)/g2)}({n*h2}-{n}{a2}({b2})/{c2}-{n}{d2}({f2})/{g2}))/{{{p}/(({m}*d)/g1)}{m}*{a1}/{c1}+{p/((n*d)/g2)}{n}*{a2}/{c2}}+{m}{d1}(y)/{g1}} &= \\simplify{{m*h1}-{m}{a1}({b1})/{c1}-{m}{d1}({f1})/{g1}}
\\end{align}

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\\begin{align}
\\simplify{y} &=\\simplify[std]{{((a1*a2*b1*g1*g2-a1*a2*b2*g1*g2+a2*c1*d1*f1*g2-a1*c2*d2*f2*g1-a2*c1*g1*g2*h1+a1*c2*g1*g2*h2)/(a1*c2*d2*g1-a2*c1*d1*g2))}}
\\end{align}

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Shows how to define variables to stop degenerate examples.

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