document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=57068>Question 57068</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Question:<BR>\n" );
document.write( "Create systems of linear equations with each possible number of solutions.\r<BR>\n" );
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document.write( "1. Consistent System (or one solution):\r<BR>\n" );
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document.write( "2. Inconsistent Graph (or no solutions):\r<BR>\n" );
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document.write( "3. Dependent Equation (or infinitely many solutions):\r<BR>\n" );
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document.write( "I know an example is required here, but how do you figure it out?  I'm not used to making my own problems up, any help would be appreciated.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #38774 by </B><A HREF=/tutors/aboutme.mpl?userid=funmath><B>funmath(2933)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=funmath><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=38774\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Question:<BR>\n" );
document.write( "Create systems of linear equations with each possible number of solutions.\r<BR>\n" );
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document.write( "1. Consistent System (or one solution):<BR>\n" );
document.write( "These two lines intersect at exactly one point.<BR>\n" );
document.write( "A consistent system would include two lines with different slopes.<BR>\n" );
document.write( "The slope intercept form of a line is:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=mx%2Bb%29 ALIGN=MIDDLE  ALT=\"highlight%28y=mx%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where m=slope and (0,b)=y-intercept.<BR>\n" );
document.write( "In a consistent system, the slope of L1) would not equal the slope of L2).  (m1 not=m2) <BR>\n" );
document.write( "Ex.<BR>\n" );
document.write( "L1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%2B4 ALIGN=MIDDLE  ALT=\"y=2x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "L2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x%2B5 ALIGN=MIDDLE  ALT=\"y=3x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "m1=2 and m2=3<BR>\n" );
document.write( "2 does not = 3<BR>\n" );
document.write( "2. Inconsistent Graph (or no solutions):<BR>\n" );
document.write( "These two lines are parallel and never intersect.<BR>\n" );
document.write( "m1=m2, but (0,b1) does not  = (0,b2)<BR>\n" );
document.write( "Ex.<BR>\n" );
document.write( "L1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%2B4 ALIGN=MIDDLE  ALT=\"y=2x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "L2)  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x%2B5 ALIGN=MIDDLE  ALT=\"y=2x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "m1=2 and m2=2, 2=2.<BR>\n" );
document.write( "but b1=4 and b2=5, 4 not=5<BR>\n" );
document.write( "Notice that you'd have to either have these lines in slope intercept form to recognize this, or if you tried to solve it, your variables would be eliminated and you'd be left with two numbers that are not equal equal to each other.  In this case (4=5) or (0=1), depending on the method you use.\r<BR>\n" );
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document.write( "3. Dependent Equation (or infinitely many solutions):<BR>\n" );
document.write( "These two lines are lying right on top of each other and have all points in common.<BR>\n" );
document.write( "m1=m2 and b1=b2<BR>\n" );
document.write( "L1)y=2x+4<BR>\n" );
document.write( "L2)y=2x+4<BR>\n" );
document.write( "m1=2 and m2=2, 2=2<BR>\n" );
document.write( "b1=4 and b2=4, 4=4<BR>\n" );
document.write( "This is obvious when the lines are in slope intercept form and simplified, but:<BR>\n" );
document.write( "These same lines can be written as:<BR>\n" );
document.write( "L1) -2x+y=4<BR>\n" );
document.write( "L2) -4x+2y=8<BR>\n" );
document.write( "If you tried to solve it, your variables would be eliminated and you'd get 0=0, or 4=4, something that ='s itself.<BR>\n" );
document.write( "I hope that helps you to pick your own lines.<BR>\n" );
document.write( "Happy Calculating!!!<BR>\n" );
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