document.write( "Question 572522: solve the following system by graphing and algebraically.
\n" ); document.write( "y=1-x^2
\n" ); document.write( "4x^2+y^2=16 \n" ); document.write( "

Algebra.Com's Answer #368432 by KMST(5328) $\"\"$ $\"About$

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ALGEBRAICALLY
\n" ); document.write( " $\"y=1-x%5E2\"$ --> $\"x%5E2=1-y\"$
\n" ); document.write( "Substituting into $\"4x%5E2%2By%5E2=16\"$ ,
\n" ); document.write( " $\"4%281-y%29%2By%5E2=16\"$ --> $\"4-4y%2By%5E2=16\"$ --> $\"y%5E2-4y-12=0\"$
\n" ); document.write( "The solutions to the above equation can be found by factoring
\n" ); document.write( " $\"y%5E2-4y-12=0\"$ --> $\"%28y-6%29%28y%2B2%29=0\"$ --> $\"y=6\"$ or $\"y=-2\"$
\n" ); document.write( " $\"y=6\"$ does not yield a real solution for x in $\"x%5E2=1-y\"$
\n" ); document.write( " $\"y=-2\"$ results in $\"x%5E2=1-%28-2%29\"$ --> $\"x%5E2=3\"$ and $\"x=sqrt%283%29\"$ or $\"x=-sqrt%283%29\"$
\n" ); document.write( "The solutions are ( $\"-sqrt%283%29\"$ ,-2) and ( $\"sqrt%283%29\"$ ,-2).
\n" ); document.write( "GRAPHICALLY
\n" ); document.write( "I do not know if use of a graphing calculator is expected, but without it we could sketch, and then calculate to see if what looks like the intersection of the graphs really adds up.
\n" ); document.write( " $\"y=1-x%5E2\"$ graphs as a parabola with maximum at (0,1) and x-intercepts at x=-1 and x=1.
\n" ); document.write( " $\"4x%5E2%2By%5E2=16\"$ dividing both sides by 16 turns into $\"x%5E2%2F4%2By%5E2%2F16=1\"$ .
\n" ); document.write( "That's the standard form of an ellipse centered at the origin, y intercepts (vertices) at y=-4 and y=4, and x-intercepts at x=-2 and x=2.
\n" ); document.write( "That would make us expect that the curves cross at negative values of y, with x between -2 and 2.
\n" ); document.write( "The graphs would look like this
\n" ); document.write( "

If the graph suggested to you that the curves cross about at about y=-2, calculations would show you that y=-2 gives the same x value for both equations and you would have the solution. \n" ); document.write( "

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