document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=169955>Question 169955</A>: <I><SPAN STYLE=\"word-break: break-all;\"> By looking at two linear equations, is there a way you can tell if the corresponding lines are parallel? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #125421 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=125421\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> That depends a great deal on the complexity (or lack thereof) of the equations and your own mental calculation skills.\r<BR>\n" );
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document.write( "First of all, I have to presume that you are dealing with two-variable equations.\r<BR>\n" );
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document.write( "Given that, if they are in slope-intercept form (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx%2Bb ALIGN=MIDDLE  ALT=\"y=mx%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >), determining parallelism is very easy.  If the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > in both equations is the same and the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > is different, you have two parallel lines.  If the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also the same in both equations, you have two representations of the same line.  If the two <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m ALIGN=MIDDLE  ALT=\"m\"   DISABLED_style= \"max-width:90%; height:auto;\" > values are different, then the lines are not parallel.\r<BR>\n" );
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document.write( "If the equations are in standard form (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ax%2BBy=C ALIGN=MIDDLE  ALT=\"Ax%2BBy=C\"   DISABLED_style= \"max-width:90%; height:auto;\" >), you have to do a little mental arithmetic.  Look at the fraction <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2FB ALIGN=MIDDLE  ALT=\"A%2FB\"   DISABLED_style= \"max-width:90%; height:auto;\" > reduced to lowest terms for each of the equations.  If different, then the lines are not parallel.  If the same, you have to look at the fraction <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2FB ALIGN=MIDDLE  ALT=\"C%2FB\"   DISABLED_style= \"max-width:90%; height:auto;\" > reduced to lowest terms for each equation to determine if the lines are parallel or the same line.\r<BR>\n" );
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document.write( "Examples:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B4y=6 ALIGN=MIDDLE  ALT=\"2x%2B4y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y=3 ALIGN=MIDDLE  ALT=\"x%2B2y=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2FB ALIGN=MIDDLE  ALT=\"A%2FB\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the first equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F4 ALIGN=MIDDLE  ALT=\"2%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > which reduces to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2FB ALIGN=MIDDLE  ALT=\"A%2FB\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the second equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "They are the same, so you potentially have parallel lines.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2FB ALIGN=MIDDLE  ALT=\"C%2FB\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the first equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F4 ALIGN=MIDDLE  ALT=\"6%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > which reduces to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2 ALIGN=MIDDLE  ALT=\"3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2FB ALIGN=MIDDLE  ALT=\"C%2FB\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the second equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2 ALIGN=MIDDLE  ALT=\"3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "They are also the same so the lines are the same line, not parallel lines\r<BR>\n" );
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document.write( "On the other hand if you had started with:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B4y=6 ALIGN=MIDDLE  ALT=\"2x%2B4y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y=4 ALIGN=MIDDLE  ALT=\"x%2B2y=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2FB ALIGN=MIDDLE  ALT=\"A%2FB\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the first equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F4 ALIGN=MIDDLE  ALT=\"2%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > which reduces to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2FB ALIGN=MIDDLE  ALT=\"A%2FB\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the second equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "They are the same, so you potentially have parallel lines.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2FB ALIGN=MIDDLE  ALT=\"C%2FB\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the first equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2F4 ALIGN=MIDDLE  ALT=\"6%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > which reduces to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F2 ALIGN=MIDDLE  ALT=\"3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2FB ALIGN=MIDDLE  ALT=\"C%2FB\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the second equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F2 ALIGN=MIDDLE  ALT=\"4%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which reduces to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "They are NOT the same, so you do have parallel lines in this case.\r<BR>\n" );
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document.write( "For the third case, presume you start with:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B4y=6 ALIGN=MIDDLE  ALT=\"2x%2B4y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2By=4 ALIGN=MIDDLE  ALT=\"2x%2By=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2FB ALIGN=MIDDLE  ALT=\"A%2FB\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the first equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F4 ALIGN=MIDDLE  ALT=\"2%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > which reduces to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2FB ALIGN=MIDDLE  ALT=\"A%2FB\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the second equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F1 ALIGN=MIDDLE  ALT=\"2%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which reduces to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "They are NOT the same, so the lines are not parallel and intersect in exactly one point.\r<BR>\n" );
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document.write( "The examples given were very simple in terms of the character of the coefficients.  Large number coefficients, fractional coefficients (decimal or otherwise), and irrational coefficients may stretch the ability of some folks to do the calculations mentally.  The point is that virtually no one would be able to determine parallelism for ANY pair of two-variable linear equations, but for the types of problems that are presented in a typical first-year Algebra course, the above method should suffice in the vast majority of cases.\r<BR>\n" );
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document.write( "Of course, once you add the complexity of a third variable, all bets are off.  In the first place, you now have four possiblities:  The lines can be parallel, they can be the same line, they can be 'skew' lines (the lines themselves are not parallel, but lie in parallel planes), or they just might intersect in a single point.  Four or more variables, the idea of parallelism no longer applies because you can't draw a picture in four or more dimensions.\r<BR>\n" );
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document.write( "Hope this helps.<BR>\n" );
document.write( "John </SPAN>\n" );
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