document.write( "Question 1182579: Solve the system by using Elimination Method:
\n" ); document.write( "1. 3x² + y² = 21
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Algebra.Com's Answer #812655 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "Solve the system by using Elimination Method:
\n" ); document.write( "1. 3x² + y² = 21
\n" ); document.write( "4x² - 2y² = -2
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document.write( "Your starting equations are\r\n" );
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document.write( "    3x^2 +  y^2 = 21      (1)\r\n" );
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document.write( "    4x^2 - 2y^2 = -2      (2)\r\n" );
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document.write( "Multiply equation (1) by 2 (both sides).   Keep equation (2) as is\r\n" );
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document.write( "    6x^2 + 2y^2 = 42      (1')\r\n" );
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document.write( "    4x^2 - 2y^2 = -2      (2')\r\n" );
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document.write( "Now add equations (1') and (2').  The terms \" 2y^2 \"  and  \" -2y^2 \"  will cancel each other (elimination),\r\n" );
document.write( "and you will get single equation for one unknown x, only:\r\n" );
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document.write( "    6x^2 + 4x^2 = 42 + (-2)\r\n" );
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document.write( "        10x^2   = 40\r\n" );
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document.write( "          x^2   = 40/10 = 4\r\n" );
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document.write( "          x             =  = +/- 2.\r\n" );
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document.write( "For now, we have two solutions for x:  +2  and -2.\r\n" );
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document.write( "Substitute x= 2 into equation (1).  You wil get then\r\n" );
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document.write( "    3*2^2 + y^2 = 21  --->  3*4 + y^2 = 21  --->  y^2 = 21 - 12 = 9  --->  y =  = +/- 3\r\n" );
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document.write( "So, with x= 2, you have two solutions  (x,y) = (2,3)  and  (x,y) = (2,-3).\r\n" );
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document.write( "Next, substitute x= -2 into equation (1).  You wil get then\r\n" );
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document.write( "    3*(-2)^2 + y^2 = 21  --->  3*4 + y^2 = 21  --->  y^2 = 21 - 12 = 9  --->  y =  = +/- 3.\r\n" );
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document.write( "So, with x= -2, you have two solutions  (x,y) = (-2,3)  and  (x,y) = (-2,-3).\r\n" );
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document.write( "Thus you get 4 (four) different pairs solutions for the given equations.\r\n" );
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document.write( "ANSWER.  The given system has 4 (four) solutions  (x,y) = (2,3), (2,-3), (-2,3)  and  (-2,-3).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved and explained in all details.\r
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\n" ); document.write( "\n" ); document.write( "First equation of the given system represents an ellipse.\r
\n" ); document.write( "\n" ); document.write( "Second equation of the given system represents a hyperbola, having two separate branches.\r
\n" ); document.write( "\n" ); document.write( "Four solutions represent four intersection points of these figures.\r
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\n" ); document.write( "\n" ); document.write( "If you want to see many other similar (and different) problems solved, look into the lesson\r
\n" ); document.write( "\n" ); document.write( " - Solving systems of algebraic equations of degree 2 \r
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\n" ); document.write( "\n" ); document.write( "Happy learning (!)\r
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