document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=934343>Question 934343</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The coordinates below represent two linear equations. <BR>\n" );
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document.write( "How many solutions does this system of equations have?<BR>\n" );
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document.write( "x	y<BR>\n" );
document.write( "–6	3<BR>\n" );
document.write( "3	6<BR>\n" );
document.write( "Line 2<BR>\n" );
document.write( "x	y<BR>\n" );
document.write( "–3	1<BR>\n" );
document.write( "3	3<BR>\n" );
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document.write( " A.<BR>\n" );
document.write( "0<BR>\n" );
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document.write( " B.<BR>\n" );
document.write( "exactly 1<BR>\n" );
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document.write( " C.<BR>\n" );
document.write( "exactly 2<BR>\n" );
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document.write( " D.<BR>\n" );
document.write( "infinitely many </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #568176 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=568176\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> given:<BR>\n" );
document.write( "Line 1<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >|	<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" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >|	<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >|	<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "first find equation of a line passing through given points:\r<BR>\n" );
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document.write( " <TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=change_this_name10094.solver>Find the equation of line going through points</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>hahaWe are trying to find equation of form y=ax+b, where a is slope, and b is intercept, which passes through points (x1, y1) = (-6, 3) and (x2, y2) = (3, 6).<BR>\n" );
document.write( "  Slope a is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29+=+%286-3%29%2F%283--6%29+=+0.333333333333333 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  Intercept is found from equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Ax%5B1%5D%2Bb+=+y%5B1%5D ALIGN=MIDDLE  ALT=\"a%2Ax%5B1%5D%2Bb+=+y%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.333333333333333%2A-6+%2Bb+=+5 ALIGN=MIDDLE  ALT=\"0.333333333333333%2A-6+%2Bb+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. From that, <BR>\n" );
document.write( "  intercept b is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=y%5B1%5D-a%2Ax%5B1%5D ALIGN=MIDDLE  ALT=\"b=y%5B1%5D-a%2Ax%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=3-0.333333333333333%2A-6+=+5 ALIGN=MIDDLE  ALT=\"b=3-0.333333333333333%2A-6+=+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  y=(0.333333333333333)x + (5)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Your graph:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A++drawing%28600%2C+600%2C+-30%2C+30%2C+-30%2C+30%2C%0D%0A++++graph%28+600%2C+600%2C+-30%2C+30%2C+-30%2C+30%2C+0.333333333333333%2Ax%2B5+%29%2C%0D%0A++++red%28+locate%28+-6%2C+3%2C+%22-6%2C+3%22+%29%2C+circle%28+-6%2C+3%2C+0.15+%29+%29%2C%0D%0A++++green%28+locate%28+3%2C+6%2C+%223%2C+6%22+%29%2C+circle%28+3%2C+6%2C+0.15+%29+%29%0D%0A++%29%0D%0A++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( " \r<BR>\n" );
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document.write( "Line 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >|	<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" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >|	<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >|	<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=change_this_name10094.solver>Find the equation of line going through points</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>hahaWe are trying to find equation of form y=ax+b, where a is slope, and b is intercept, which passes through points (x1, y1) = (-3, 1) and (x2, y2) = (3, 3).<BR>\n" );
document.write( "  Slope a is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29+=+%283-1%29%2F%283--3%29+=+0.333333333333333 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  Intercept is found from equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Ax%5B1%5D%2Bb+=+y%5B1%5D ALIGN=MIDDLE  ALT=\"a%2Ax%5B1%5D%2Bb+=+y%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.333333333333333%2A-3+%2Bb+=+2 ALIGN=MIDDLE  ALT=\"0.333333333333333%2A-3+%2Bb+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. From that, <BR>\n" );
document.write( "  intercept b is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=y%5B1%5D-a%2Ax%5B1%5D ALIGN=MIDDLE  ALT=\"b=y%5B1%5D-a%2Ax%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=1-0.333333333333333%2A-3+=+2 ALIGN=MIDDLE  ALT=\"b=1-0.333333333333333%2A-3+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  y=(0.333333333333333)x + (2)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Your graph:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A++drawing%28600%2C+600%2C+-30%2C+30%2C+-30%2C+30%2C%0D%0A++++graph%28+600%2C+600%2C+-30%2C+30%2C+-30%2C+30%2C+0.333333333333333%2Ax%2B2+%29%2C%0D%0A++++red%28+locate%28+-3%2C+1%2C+%22-3%2C+1%22+%29%2C+circle%28+-3%2C+1%2C+0.15+%29+%29%2C%0D%0A++++green%28+locate%28+3%2C+3%2C+%223%2C+3%22+%29%2C+circle%28+3%2C+3%2C+0.15+%29+%29%0D%0A++%29%0D%0A++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( " \r<BR>\n" );
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document.write( "see both lines on a graph:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%280.333333333333333%29x+%2B+2%2C+%280.333333333333333%29x+%2B+5%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From the graph, we can see that the two lines are parallel and will never intersect. So there are no solutions and the system is inconsistent. \r<BR>\n" );
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document.write( " according to Euclidean geometry, if two lines are distinct but have the same slope they are said to be parallel and have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=no ALIGN=MIDDLE  ALT=\"no\"   DISABLED_style= \"max-width:90%; height:auto;\" > points in common.\r<BR>\n" );
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document.write( "so, your answer is A. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "It is also  good to know that, according to non-Euclidean geometry (so called projective geometry) any pair of lines <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=always ALIGN=MIDDLE  ALT=\"always\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+intersects ALIGN=MIDDLE  ALT=\"+intersects\"   DISABLED_style= \"max-width:90%; height:auto;\" > at some point, but parallel lines do not intersect in the real plane. The line at infinity is added to the real plane. This completes the plane, because now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=parallel ALIGN=MIDDLE  ALT=\"parallel\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+lines ALIGN=MIDDLE  ALT=\"+lines\"   DISABLED_style= \"max-width:90%; height:auto;\" > intersect at a point which lies on the line at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Also, if any pair of lines intersects at a point on the line at infinity, then the pair of lines is parallel.\r<BR>\n" );
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document.write( " but we will keep A.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > as an answer to your question<BR>\n" );
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