Ingrese su respuesta como números enteros o bien fracciones.
\n$x=$ [[0]]
\n$y=$ [[1]]
\n", "marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "gaps": [{"checkingtype": "absdiff", "variableReplacementStrategy": "originalfirst", "marks": 1, "showCorrectAnswer": true, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "notallowed": {"strings": ["."], "showStrings": false, "message": "Input your answer as a fraction and not a decimal.
", "partialCredit": 0}, "answer": "{-((a1*b1*c2*d2*g1-a2*b2*c1*d1*g2+c1*c2*d1*d2*f1-c1*c2*d1*d2*f2-c1*c2*d2*g1*h1+c1*c2*d1*g2*h2)/(a1*c2*d2*g1-a2*c1*d1*g2))}", "vsetrangepoints": 5, "vsetrange": [0, 1], "answersimplification": "Std", "expectedvariablenames": [], "checkvariablenames": false}, {"checkingtype": "absdiff", "variableReplacementStrategy": "originalfirst", "marks": 1, "showCorrectAnswer": true, "variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "notallowed": {"strings": ["."], "showStrings": false, "message": "Input your answer as a fraction and not as a decimal.
", "partialCredit": 0}, "answer": "{((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))}", "vsetrangepoints": 5, "vsetrange": [0, 1], "answersimplification": "Std", "expectedvariablenames": [], "checkvariablenames": false}]}], "statement": "Resolver el siguiente sistema de Ecuaciones:
\n\\[\\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*}\\]
\n", "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {"std": ["All", "fractionnumbers"]}, "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Shows how to define variables to stop degenerate examples.
"}, "name": "Sistema de Ecuaciones (2x2) (2)", "variables": {"h2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-2..2 except 0)", "description": "", "name": "h2"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except 0)", "description": "", "name": "b"}, "p": {"group": "Unnamed group", "templateType": "anything", "definition": "lcm((m*d)/g1,(n*d)/g2)", "description": "", "name": "p"}, "f2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-3..3 except 0)", "description": "", "name": "f2"}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-3,-2,2,3)", "description": "", "name": "d1"}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..3)", "description": "", "name": "a1"}, "d2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "-d1", "description": "", "name": "d2"}, "c2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random([2,3,4,6] except [a2,2a2,3a2])", "description": "", "name": "c2"}, "f1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-2..2)", "description": "", "name": "f1"}, "a2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2,3)", "description": "", "name": "a2"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..6 except [5,a1,2a1,3a1])", "description": "", "name": "c1"}, "m": {"group": "Unnamed group", "templateType": "anything", "definition": "lcm(c1,g1)", "description": "", "name": "m"}, "g2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..5 except [-d2,-2d2,d2,2d2,c2])", "description": "", "name": "g2"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9 except 0)", "description": "", "name": "c"}, "g1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(2..5 except [d1,2d1,-d1,-2d1])", "description": "", "name": "g1"}, "b2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-2..2)", "description": "", "name": "b2"}, "n": {"group": "Unnamed group", "templateType": "anything", "definition": "lcm(c2,g2)", "description": "", "name": "n"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "abs(d1)", "description": "", "name": "d"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "a"}, "h1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-3..3)", "description": "", "name": "h1"}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-3..3 except 0)", "description": "", "name": "b1"}}, "extensions": [], "functions": {}, "advice": "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:
\n\\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}
Multiplicamos y simplificamos:
\n\\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}
Volvemos a multiplicar, reducimos y ordenamos:
\n\\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}
Ahora multiplicamos ambas ecuaciones por cantidades adecuadas de modo de aplicar el método de reducción.
\nUna 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:
\n\\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}
Sumando ambas ecuaciones:
\n\\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}
Resolvemos para obtener el valor de $x$
\n \\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}
Finalmente reemplazamos el valor de $x$ en la ecuación $\\,(*)\\,$ y así obtenemos el valor de $y$:
\n \\begin{align}
\\simplify{{m}{a1}(x)/{c1}+{m}{d1}(y)/{g1}} &= \\simplify{{m*h1}-{m}{a1}({b1})/{c1}-{m}{d1}({f1})/{g1}}
\\end{align}
\\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}
\\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}