2007-12-07T10:15:14Z https://www.matematicasies.com/Sistema-2x2-No-Lineal,1522#comment266 2007-12-07T10:15:14Z

<p>En el sistema<br></p> <p><img src='https://www.matematicasies.com/local/cache-TeX/9fbf2fc312875aceb4541865d3bcb37f.png' style="vertical-align:middle;" width="162" height="72" alt="\left\{ \begin{array}{lll} y^2 - 2y + 1 = x \\ \sqrt{x} + y = 5 \end{array} \right." title="\left\{ \begin{array}{lll} y^2 - 2y + 1 = x \\ \sqrt{x} + y = 5 \end{array} \right."></p> <p>Tenemos despejada <img src='https://www.matematicasies.com/local/cache-TeX/9dd4e461268c8034f5c8564e155c67a6.png' style="vertical-align:middle;" width="17" height="30" alt="x" title="x">. Resolvamos por sustitución (sustituimos x por su valor en la segunda ecuación):</p> <p><img src='https://www.matematicasies.com/local/cache-TeX/0ce17b2921f5a344d9772676c9120436.png' style="vertical-align:middle;" width="187" height="58" alt="\sqrt{y^2 - 2y + 1} + y = 5" title="\sqrt{y^2 - 2y + 1} + y = 5"></p> <p>Podemos resolver como una ecuación irracional (aislando la raíz y elevando al cuadrado).</p> <p>Sin embargo hay un camino más directo .. teniendo en cuenta que:<br> <img src='https://www.matematicasies.com/local/cache-TeX/25b97d510e5e7f53eab4a40f97ee4a2e.png' style="vertical-align:middle;" width="193" height="47" alt="y^2 - 2y + 1 = (y-1)^2" title="y^2 - 2y + 1 = (y-1)^2"></p> <p><img src='https://www.matematicasies.com/local/cache-TeX/0ce17b2921f5a344d9772676c9120436.png' style="vertical-align:middle;" width="187" height="58" alt="\sqrt{y^2 - 2y + 1} + y = 5" title="\sqrt{y^2 - 2y + 1} + y = 5"><br> <img src='https://www.matematicasies.com/local/cache-TeX/1dca534bdc458748094a1f6615583342.png' style="vertical-align:middle;" width="157" height="57" alt="\sqrt{(y-1)^2} + y = 5" title="\sqrt{(y-1)^2} + y = 5"><br> Simplificamos cuadrado con raíz<br> <img src='https://www.matematicasies.com/local/cache-TeX/c87c60c569fef7d4d261a8c7eb0dd72b.png' style="vertical-align:middle;" width="120" height="38" alt="y-1 +y=5" title="y-1 +y=5"><br> <img src='https://www.matematicasies.com/local/cache-TeX/d44939440a2ce1f06e92a47d4f0f7d7c.png' style="vertical-align:middle;" width="60" height="38" alt="2y = 6" title="2y = 6"> <br> <img src='https://www.matematicasies.com/local/cache-TeX/8f9c515f7fdd5fe9424cb404874ed200.png' style="vertical-align:middle;" width="50" height="38" alt="y = 3" title="y = 3"></p> <p>Ahora calculamos <img src='https://www.matematicasies.com/local/cache-TeX/9dd4e461268c8034f5c8564e155c67a6.png' style="vertical-align:middle;" width="17" height="30" alt="x" title="x"> que la teníamos despejada:<br></p> <p><img src='https://www.matematicasies.com/local/cache-TeX/98906f6ebf77710c97631adbef5a2b7d.png' style="vertical-align:middle;" width="140" height="47" alt="y^2 - 2y + 1 = x" title="y^2 - 2y + 1 = x"><br> <img src='https://www.matematicasies.com/local/cache-TeX/2263498e46d4e3a76063d16ef9b930b2.png' style="vertical-align:middle;" width="153" height="47" alt="3^2 - 2 \cdot 3 + 1 = x" title="3^2 - 2 \cdot 3 + 1 = x"><br> <img src='https://www.matematicasies.com/local/cache-TeX/a133180e146cb9b12f9e9d65bffe3ee3.png' style="vertical-align:middle;" width="120" height="38" alt="9 - 6 + 1 = x" title="9 - 6 + 1 = x"><br> <img src='https://www.matematicasies.com/local/cache-TeX/440bcb2225cd249b09bb29454f83249d.png' style="vertical-align:middle;" width="50" height="38" alt="x=4" title="x=4"></p> <p>Finalmente comprobamos que las soluciones verifican el sistema:</p> <p><img src='https://www.matematicasies.com/local/cache-TeX/a37a7d435200666ff0c6c538cf299e01.png' style="vertical-align:middle;" width="170" height="72" alt="\left\{ \begin{array}{lll} 3^2 - 2 \cdot 3 + 1 = 4 \\ \sqrt{4} + 3 = 5 \end{array} \right." title="\left\{ \begin{array}{lll} 3^2 - 2 \cdot 3 + 1 = 4 \\ \sqrt{4} + 3 = 5 \end{array} \right."></p> <p>Por tanto las soluciones son <img src='https://www.matematicasies.com/local/cache-TeX/440bcb2225cd249b09bb29454f83249d.png' style="vertical-align:middle;" width="50" height="38" alt="x=4" title="x=4"> , <img src='https://www.matematicasies.com/local/cache-TeX/2afffe4b624abdd27735b7626f7a810d.png' style="vertical-align:middle;" width="50" height="38" alt="y=3" title="y=3"><br class="autobr"> </p>