document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=709430>Question 709430</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine whether each system of linear equation has one and only one sol. Infinite many or no sol. \r<BR>\n" );
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document.write( "3/2x-2y=4 and x+1/3y=2 couldu help me with this problem I am lost  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #436576 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=436576\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Because that system has fractional coefficients,<BR>\n" );
document.write( "my first step would be to \"eliminate denominators\" in each equation,<BR>\n" );
document.write( "by multiplying both sides of the equal sign times an appropriate number.<BR>\n" );
document.write( "I would multiply times 2 for the first equation<BR>\n" );
document.write( "and times 3 for the second equation<BR>\n" );
document.write( "to transform the whole system into one that is easier for me.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28%283%2F2%29%2Ax-2y=4%2Cx%2B%281%2F3%29%2Ay=2%29 ALIGN=MIDDLE  ALT=\"system%28%283%2F2%29%2Ax-2y=4%2Cx%2B%281%2F3%29%2Ay=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x-4y=8%2C3x%2By=6%29 ALIGN=MIDDLE  ALT=\"system%283x-4y=8%2C3x%2By=6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "That system has one and only one solution.<BR>\n" );
document.write( "Why? Because the ratios of coefficients of x and y are different.<BR>\n" );
document.write( "Let me explain by example<BR>\n" );
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document.write( "Obviously <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x%2By=6%2C3x%2By=6%29 ALIGN=MIDDLE  ALT=\"system%283x%2By=6%2C3x%2By=6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > has many solutions<BR>\n" );
document.write( "because both equations are the same,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2By=6%29 ALIGN=MIDDLE  ALT=\"3x%2By=6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents one line<BR>\n" );
document.write( "with an infinite number of (x,y) points that are solutions to that equation<BR>\n" );
document.write( "and solutions to the system.<BR>\n" );
document.write( "That system could appear \"in disguise\" and it would not be so obvious, as in<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28%283%2F2%29x%2B%281%2F2%29y=3%2Cx%2B%281%2F3%29y=2%29 ALIGN=MIDDLE  ALT=\"system%28%283%2F2%29x%2B%281%2F2%29y=3%2Cx%2B%281%2F3%29y=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%286x%2B2y=12%2C3x%2By=6%29 ALIGN=MIDDLE  ALT=\"system%286x%2B2y=12%2C3x%2By=6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "All those equations are equivalent, because one can be obtained from another one<BR>\n" );
document.write( "by multiplying both sides of the equal sign times an appropriate number.<BR>\n" );
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document.write( "On the other hand, the system <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x%2By=6%2C3x%2By=5%29 ALIGN=MIDDLE  ALT=\"system%283x%2By=6%2C3x%2By=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "obviously has no solutions.<BR>\n" );
document.write( "It could also be disguised to make it not so obvious.<BR>\n" );
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document.write( "Once your system was transformed into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x-4y=8%2C3x%2By=6%29 ALIGN=MIDDLE  ALT=\"system%283x-4y=8%2C3x%2By=6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "It was clear that it did not fit into the no-solution or infinite-solutions situations described above,<BR>\n" );
document.write( "because both equations had 3 as the coefficient for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > but had different coefficients for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "The infinite number (x,y) data pairs that are solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2By=6 ALIGN=MIDDLE  ALT=\"3x%2By=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "represent the points in the straight line that is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2By=6 ALIGN=MIDDLE  ALT=\"3x%2By=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The infinite number (x,y) data pairs that are solutions to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-4y=8 ALIGN=MIDDLE  ALT=\"3x-4y=8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "represent the points in the straight line that is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-4y=8 ALIGN=MIDDLE  ALT=\"3x-4y=8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The two lines intersect at just one common point,<BR>\n" );
document.write( "and the coordinates of that point constitute the solution to the system <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x-4y=8%2C3x%2By=6%29 ALIGN=MIDDLE  ALT=\"system%283x-4y=8%2C3x%2By=6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In this case, that point is (32/15, -2/5) and the solution to the system is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=32%2F15%29 ALIGN=MIDDLE  ALT=\"highlight%28x=32%2F15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=-2%2F5%29 ALIGN=MIDDLE  ALT=\"highlight%28y=-2%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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