document.write( "Question 1173777: x^2-2x-4y^2+8y-2=0 and 5x^2-10x+12y^2+24y-58=0 system of nonlinear equation \n" ); document.write( "

Algebra.Com's Answer #799088 by ikleyn(52812) $\"\"$ $\"About$

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document.write( " x^2 -  2x - 4y^2  +  8y -  2 = 0    (1)\r\n" );
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document.write( "5x^2 - 10x + 12y^2 + 24y - 58 = 0    (2)\r\n" );
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document.write( "Look at the system. You see that the two terms with  x^2 and x  of the second equation are five times \r\n" );
document.write( "the corresponding terms of the first equation.\r\n" );
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document.write( "It gives me an idea to multiply first equation by 5 and then subtract from the second equation\r\n" );
document.write( "to ELIMINATE all the terms with x.\r\n" );
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document.write( "So my first step is to multiply first equation by 5, while keeping second equation as is\r\n" );
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document.write( "    5x^2 - 10x - 20y^2 + 40y - 10 = 0     (1')\r\n" );
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document.write( "    5x^2 - 10x + 12y^2 + 24y - 58 = 0     (2)\r\n" );
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document.write( "My second step is to subtract equation (1') from equation (2).  In this way, I get\r\n" );
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document.write( "                 32y^2 - 16y - 48 = 0.\r\n" );
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document.write( "Next, use the quadratic formula\r\n" );
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document.write( "     =  =  = .\r\n" );
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document.write( "So, I have two solutions for y:   = -1  and   =  = .\r\n" );
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document.write( "The last step is to substitute these found values of y into equation (1) and solve it for x.\r\n" );
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document.write( "It is just fully mechanical job and I leave it to you.\r\n" );
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document.write( "You should find two values of x for each of the two values of y.\r\n" );
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document.write( "It will give you four final solutions for pairs (x,y) at the end.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Having so clear instructions, boldly go forward and complete the solution on your own.\r
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\n" ); document.write( "\n" ); document.write( "By the way, first equation represents a hyperbola, while the second equation represents an ellipse.\r
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\n" ); document.write( "\n" ); document.write( "The solution represents 4 (four) intersection points.\r
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\n" ); document.write( "\n" ); document.write( "It is good if you understand what these equations represent.\r
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\n" ); document.write( "\n" ); document.write( "It is not necessary for formal solution at this level, but nevertheless . . . \r
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