document.write( "Question 303172: * Find the solution to this system of equations, algebraically and graphically.
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Algebra.Com's Answer #217314 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!
* Find the solution to this system of equations, algebraically and graphically.
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document.write( "First we draw the graph of the hyperbola\r\n" );
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document.write( "We put it in standard form by getting a 1 on the right\r\n" );
document.write( "by dividing every term by 16:\r\n" );
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document.write( "Then we find the semi-transverse and semi-conjugate axes \r\n" );
document.write( "and draw the defining rectangle, draw the asymptotes and \r\n" );
document.write( "sketch the hyperbola:\r\n" );
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document.write( "We do the same with the ellipse   x^2+4y^2=16:\r\n" );
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document.write( "Then we find the semi-major and semi-minor axes and sketch \r\n" );
document.write( "the hyperbola:\r\n" );
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document.write( "That gives us this graph (in green).  I'll erase the \r\n" );
document.write( "guidelines for the hyperbola and just leave the hyperbola\r\n" );
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document.write( "So we see that the only points the red hyperbola and the green\r\n" );
document.write( "ellipse have in common are the points\r\n" );
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document.write( "(x,y) = (4,0) and (x,y) = (-4,0)\r\n" );
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document.write( "So when we do the problem algebraically we should get the\r\n" );
document.write( "values of x as 4 and -4 and the value of y as 0.\r\n" );
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document.write( "Solve the first equation for :\r\n" );
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document.write( ".  Substitute  for \r\n" );
document.write( "in the second:\r\n" );
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document.write( "Substitute 0 for y in\r\n" );
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document.write( "ą.\r\n" );
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document.write( "So we do get for the solutions the same\r\n" );
document.write( "values algebraically as we saw that the points \r\n" );
document.write( "which the hyperbola and the ellipse had in common\r\n" );
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document.write( "(x,y) = (ą4,0) or the points: \r\n" );
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document.write( "(x,y) = (4,0) and (x,y) = (-4,0)        \r\n" );
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\n" ); document.write( "* Graph the solution set of this system of inequalities:\r
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document.write( "Notice that the first is just like the equation for the green\r\n" );
document.write( "ellipse in the preceding problem except that it has\r\n" );
document.write( "an  symbol instead of an equal sign .\r\n" );
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document.write( "That is the region on and inside the green ellipse with the\r\n" );
document.write( "equation .  We draw the ellipse solid \r\n" );
document.write( "not dotted because the inequality is  and not \r\n" );
document.write( ", and thus the ellipse itself is included in the\r\n" );
document.write( "graph:\r\n" );
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document.write( "Next we see the inequality .  We draw the\r\n" );
document.write( "boundary line dotted since the inequality is \r\n" );
document.write( "and not  and therefore the graph is only\r\n" );
document.write( "the area above the line and does not include the line.\r\n" );
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document.write( "So I will erase all the part that is not to be shaded and we have\r\n" );
document.write( "just the inside area. I can't shade on here, but you can on your \r\n" );
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document.write( "Edwin

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