document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=67286>Question 67286</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.The sum of two positive real numbers is 7. The sum of their squares is 26.1/2. Find the numbers.\r<BR>\n" );
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document.write( "2.The perimeter of this right-angled triangle is 60 ( the two perpendicular sides are x and y and the other side is 26).<BR>\n" );
document.write( "a) Write down two equations in x and y.<BR>\n" );
document.write( "b) Find the values of x and y, given that x>y.\r<BR>\n" );
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document.write( "3.The diagram shows two squares; the sum of their areas is 58 meter square. The sum of the lengths of their sides is 10 meter. Find the values of x and y, as shown : (the small square is with the x side and the big square is with y side. the diagram shows that x+y = 10).\r<BR>\n" );
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document.write( "4. Solve:<BR>\n" );
document.write( "   5x-2y=3<BR>\n" );
document.write( "   xsquare+ysquare-2xy=0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #47864 by </B><A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><B>ankor@dixie-net.com(22740)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><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/cgi-bin/show-face.mpl?user=ankor@dixie-net.com><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=47864\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> These problems are similar. The Method<BR>\n" );
document.write( "a) arrange the simplest equation to be used as substitution.<BR>\n" );
document.write( "b) substitute in the 2nd equation, arrange as a quadratic equation<BR>\n" );
document.write( "c) factor or solve the quadratic equation><BR>\n" );
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document.write( "1.The sum of two positive real numbers is 7.<BR>\n" );
document.write( "x + y = 7<BR>\n" );
document.write( "y = 7 - x <BR>\n" );
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document.write( "The sum of their squares is 26.1/2. Find the numbers.<BR>\n" );
document.write( "x^2 + y^2 = 26.5<BR>\n" );
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document.write( "Substitute (7-x) for y:<BR>\n" );
document.write( "x^2 + (7-x)^2 = 26.5<BR>\n" );
document.write( "x^2 + 49 - 14x + x^2 = 26.5<BR>\n" );
document.write( "x^2 + x^2 - 14x + 49 - 26.5 = 0<BR>\n" );
document.write( "2x^2 - 14x + 22.5 = 0<BR>\n" );
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document.write( "Simplify, divide by 2:<BR>\n" );
document.write( "x^2 - 7x + 11.25 = 0<BR>\n" );
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document.write( "Solving this we get:<BR>\n" );
document.write( "x = +2.5, x = +4.5 and these are to two numbers<BR>\n" );
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document.write( "2.The perimeter of this right-angled triangle is 60 ( the two perpendicular sides are x and y and the other side is 26).<BR>\n" );
document.write( "a) Write down two equations in x and y.<BR>\n" );
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document.write( "x + y + 26 = 60<BR>\n" );
document.write( "x + y = 60 - 26<BR>\n" );
document.write( "x + y = 34<BR>\n" );
document.write( "y = (34-x)<BR>\n" );
document.write( "and<BR>\n" );
document.write( "x^2 + y^2 = 26^2<BR>\n" );
document.write( "x^2 + y^2 = 676<BR>\n" );
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document.write( "b) Find the values of x and y, given that x>y.<BR>\n" );
document.write( "Substitute (34-x) for y in x^2 + y^ = 676:<BR>\n" );
document.write( "x^2 + (34-x)^2 = 676<BR>\n" );
document.write( "x^2 + 1156 - 68x + x^2 = 676<BR>\n" );
document.write( "x^2 + x^2- 68x + 1156 - 676 = 0<BR>\n" );
document.write( "2x^2 - 68x + 480 = 0 <BR>\n" );
document.write( "Simplify, divide by 2:<BR>\n" );
document.write( "x^2 - 34x + 240 = 0<BR>\n" );
document.write( "Factors to:<BR>\n" );
document.write( "(x-10)(x-24) = 0<BR>\n" );
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document.write( "In the problem x = 24, y = 10<BR>\n" );
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document.write( "3.The diagram shows two squares; the sum of their areas is 58 meter square. The sum of the lengths of their sides is 10 meter. Find the values of x and y, as shown : (the small square is with the x side and the big square is with y side. the diagram shows that x+y = 10).<BR>\n" );
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document.write( "This is the same as the previous problems,<BR>\n" );
document.write( "y = (10+x)<BR>\n" );
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document.write( "Substitute into x^2 + y^2 = 58 in the same way, solve the quadratic equation<BR>\n" );
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document.write( "4. Solve:<BR>\n" );
document.write( "5x-2y=3<BR>\n" );
document.write( "5x = 2y + 3<BR>\n" );
document.write( "x = (2/5)y + (3/5)<BR>\n" );
document.write( "x = .4y + .6 in decimals<BR>\n" );
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document.write( "x^2 + y^2 - 2xy = 0<BR>\n" );
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document.write( "Substitute (.4y + .6) for x:<BR>\n" );
document.write( "(.4y+.6)^2 + y^2 - 2y(.4y+.6) = 0<BR>\n" );
document.write( ".16y^2 + .48y + .36 + y^2 -.8y^2 - 1.2y = 0<BR>\n" );
document.write( ".16y^2 + y^2 -.8y^2 + .48y - 1.2y + .36 = 0<BR>\n" );
document.write( ".36y^2 - .72y + .36 = 0<BR>\n" );
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document.write( "Simplify, divide equation by .36, we have a simple equation:<BR>\n" );
document.write( "y^2 - 2y + 1 = 0<BR>\n" );
document.write( "Factors to:<BR>\n" );
document.write( "(y - 1)(y - 1) = 0<BR>\n" );
document.write( "y = +1<BR>\n" );
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document.write( "Find x:<BR>\n" );
document.write( "5x - 2y = 3<BR>\n" );
document.write( "5x - 2(1) = 3<BR>\n" );
document.write( "5x = 3 + 2<BR>\n" );
document.write( "x = 5/5<BR>\n" );
document.write( "x = 1<BR>\n" );
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document.write( "Check: 1^2 + 1^2 - 2(1*1) = 0 </SPAN>\n" );
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