document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=159004>Question 159004</A>: <I><SPAN STYLE=\"word-break: break-all;\"> .. PLEASE HELP ME SOLVE THIS EQUATION.\r<BR>\n" );
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document.write( " x+y<8<BR>\n" );
document.write( "2x+y<10<BR>\n" );
document.write( "x>0,y>0\r<BR>\n" );
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document.write( "....\r<BR>\n" );
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document.write( " x+y<8       <BR>\n" );
document.write( "-x   -x     <BR>\n" );
document.write( "________ <BR>\n" );
document.write( "  Y<-x+8  \r<BR>\n" );
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document.write( "   <BR>\n" );
document.write( "   2x+y<10                <BR>\n" );
document.write( " -2x     -2x  <BR>\n" );
document.write( " ____________<BR>\n" );
document.write( "  y<-2x+10\r<BR>\n" );
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document.write( ".. i need to graph this but i do not know how to do it.!!=S<BR>\n" );
document.write( "           </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #117398 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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=117398\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the problem to be solved is:<BR>\n" );
document.write( "x+y<8 and 2*x+y<10 <BR>\n" );
document.write( "where x and y both have to be greater than 0.<BR>\n" );
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document.write( "to solve for x and y, solve for equality rather than inequality first.<BR>\n" );
document.write( "you get<BR>\n" );
document.write( "x+y=8<BR>\n" );
document.write( "2*x+y=10<BR>\n" );
document.write( "subtract second equation from first equation and you get<BR>\n" );
document.write( "x = 2<BR>\n" );
document.write( "if x = 2, then y = 6<BR>\n" );
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document.write( "for both equations to be equal with the same values of x and y in each, x must = 2 and y must = 6.<BR>\n" );
document.write( "that would be the intersection of the 2 lines on a graph.<BR>\n" );
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document.write( "going back to the original inequality for x+y to be < 8, the combination of x + y must be smaller than 8.  if x = 7, then y must be smaller than 1.  if y = 7, then x must be smaller than 1.<BR>\n" );
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document.write( "going back to the original inequality for 2*x + y < 10, the combination of 2*x + y must be smaller than 10.  if x = 4, then y must be smaller than 2.  if y = 8, then x must be smaller than 1.<BR>\n" );
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document.write( "example:<BR>\n" );
document.write( "let x = 1 and y = 2.<BR>\n" );
document.write( "x+y = 3 < 8 checks out ok.<BR>\n" );
document.write( "2*x+y = 4 < 10 checks out ok.<BR>\n" );
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document.write( "example:<BR>\n" );
document.write( "let x = 1 and y = 8<BR>\n" );
document.write( "x+y = 9 not < 8 - doesn't satisfy the equation.<BR>\n" );
document.write( "2*x + y = 10 not < 10 - doesn't satisfy the equation.<BR>\n" );
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document.write( "viewing this on a graph should help to make it clearer.<BR>\n" );
document.write( "to make the graph, we go back to the equations showing equality again.<BR>\n" );
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document.write( "first equation:<BR>\n" );
document.write( "x + y = 8<BR>\n" );
document.write( "convert to slope-intercept form (y = m*x + b)<BR>\n" );
document.write( "y = (-x) + 8<BR>\n" );
document.write( "slope is -1<BR>\n" );
document.write( "y-intercept is 8.<BR>\n" );
document.write( "when x = 0, y = 8<BR>\n" );
document.write( "when y = 0, x = 8<BR>\n" );
document.write( "we have two points that we can plot to create the graph.<BR>\n" );
document.write( "they are (0,8) and (8,0)<BR>\n" );
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document.write( "second equation:<BR>\n" );
document.write( "2*x + y = 10<BR>\n" );
document.write( "convert to slope-intercept form (y = m*x + b)<BR>\n" );
document.write( "y = (-2*x) + 10<BR>\n" );
document.write( "slope is -2<BR>\n" );
document.write( "y-intercept is 10<BR>\n" );
document.write( "when x = 0, y = 10<BR>\n" );
document.write( "when y = 0, x = 5<BR>\n" );
document.write( "we have two points that we can plot to create the graph.<BR>\n" );
document.write( "they are (0,10) and (5,0)<BR>\n" );
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document.write( "graph looks like the following.<BR>\n" );
document.write( "please scan below the graph for more comments.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28800%2C800%2C-2%2C10%2C-2%2C12%2C-x%2B8%2C-2%2Ax%2B10%29 ALIGN=MIDDLE  ALT=\"graph%28800%2C800%2C-2%2C10%2C-2%2C12%2C-x%2B8%2C-2%2Ax%2B10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "if x has to be greater than 0 and y has to be greater than 0, the answer you are looking for appears to be the area under the graph of y = (-2*x)+10 for x = 2 to 5, and the area under the graph of y = (-x)+8 for x = 0 to 2.<BR>\n" );
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document.write( "to prove this, take 2 points that don't fall within the specified areas but still fall within the areas of either curve.<BR>\n" );
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document.write( "let x = .5, and y = 8.<BR>\n" );
document.write( "then x + y < 8 becomes .5 + 8 which is not < 8 so it fails the test.<BR>\n" );
document.write( "let x = 6, and y = 1.<BR>\n" );
document.write( "then x + y < 8 becomes 6 + 1 < 8 which is ok for the first equation.<BR>\n" );
document.write( "but, ........<BR>\n" );
document.write( "2*x + y < 10 becomes 12 + 1 which is not < 10 so it fails the test.<BR>\n" );
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document.write( "since the values of x and y must satisfy both equations, if the values satisfy one equation and not the other, the test fails.<BR>\n" );
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document.write( "to prove a positive, let x = something in the specified area.<BR>\n" );
document.write( "let x = 1.9 and y = 6.<BR>\n" );
document.write( "x + y < 8 becomes 1.9 + 6 < 8 which is ok for the first equation.<BR>\n" );
document.write( "2*x + y < 10 becomes 3.8 + 6 < 10 which is ok for the second equation.<BR>\n" );
document.write( "since it passes the test for both equation, that value is good and it does fall within the specified area.<BR>\n" );
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document.write( "the specified area is ...............................<BR>\n" );
document.write( "the area under the graph of y = (-2*x)+10 for x = 2 to x = 5, and <BR>\n" );
document.write( "the area under the graph of y = (-x)+8 for x = 0 to 2.<BR>\n" );
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document.write( " </SPAN>\n" );
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