document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197265>Question 1197265</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the coordinates of the point(s) of intersection for the following linear-quadratic system algebraically:\r<BR>\n" );
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document.write( "y = x^2-7x+15 and y = 2x-5. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830472 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=830472\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The variable y is equal to both x^2-7x+15 and also to 2x-5<BR>\n" );
document.write( "Equate the right hand sides and get everything to one side.\r<BR>\n" );
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document.write( "x^2-7x+15 = 2x-5<BR>\n" );
document.write( "x^2-7x+15-2x+5 = 0<BR>\n" );
document.write( "x^2-9x+20 = 0\r<BR>\n" );
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document.write( "Then we can factor. Think of two numbers that<BR>\n" );
document.write( "A) multiply to 20, and<BR>\n" );
document.write( "B) add to -9<BR>\n" );
document.write( "Through trial and error you should arrive at -4 and -5\r<BR>\n" );
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document.write( "-4 times -5 = 20<BR>\n" );
document.write( "-4 plus -5 = -9\r<BR>\n" );
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document.write( "Therefore,<BR>\n" );
document.write( "x^2-9x+20 = 0<BR>\n" );
document.write( "(x-4)(x-5) = 0<BR>\n" );
document.write( "x-4 = 0 or x-5 = 0<BR>\n" );
document.write( "x = 4 or x = 5<BR>\n" );
document.write( "The second to last step used the zero product property\r<BR>\n" );
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document.write( "Alternative Route\r<BR>\n" );
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document.write( "Go back to x^2-9x+20 = 0<BR>\n" );
document.write( "Compare this to ax^2+bx+c = 0\r<BR>\n" );
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document.write( "We have<BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "b = -9<BR>\n" );
document.write( "c = 20<BR>\n" );
document.write( "Plug those values into the quadratic formula<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+5 ALIGN=MIDDLE  ALT=\"x+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4 ALIGN=MIDDLE  ALT=\"x+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The order of the solutions doesn't matter. \r<BR>\n" );
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document.write( "Once we've determined the x values of the solutions, we use them to find their paired y values.\r<BR>\n" );
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document.write( "Plug in x = 4<BR>\n" );
document.write( "Let's do so for the first equation<BR>\n" );
document.write( "y = x^2-7x+15<BR>\n" );
document.write( "y = 4^2-7*4+15<BR>\n" );
document.write( "y = 16-28+15<BR>\n" );
document.write( "y = -12+15<BR>\n" );
document.write( "y = 3<BR>\n" );
document.write( "Be sure to follow the order of operations PEMDAS\r<BR>\n" );
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document.write( "This basically says the input x = 4 leads to the output y = 3\r<BR>\n" );
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document.write( "Now do the same for the other equation<BR>\n" );
document.write( "y = 2x-5<BR>\n" );
document.write( "y = 2*4-5<BR>\n" );
document.write( "y = 8-5<BR>\n" );
document.write( "y = 3<BR>\n" );
document.write( "The second equation is much easier to work with, so if you only had to pick one, then I'd go for this.\r<BR>\n" );
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document.write( "However, it's good practice to check BOTH equations to verify the solution fully.\r<BR>\n" );
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document.write( "One solution is (x,y) = (4,3) which is one point where the parabola y = x^2-7x+15 and line y = 2x-5 intersect.\r<BR>\n" );
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document.write( "The other solution is (x,y) = (5,5)<BR>\n" );
document.write( "You'll plug x = 5 into either equation to find that y = 5 pairs up with it. \r<BR>\n" );
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document.write( "Answers: <font color=red>(4,3) and (5,5)</font>\r<BR>\n" );
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document.write( "Visual Verification<BR>\n" );
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document.write( "I recommend using Desmos or GeoGebra as graphing tools to verify the answer. <BR>\n" );
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