document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210619>Question 1210619</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Fill in the blanks so that the resulting system does not have a unique solution.\r<BR>\n" );
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document.write( "8x + 12y = 26\r<BR>\n" );
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document.write( "8x + 12y = 1\r<BR>\n" );
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document.write( "14x + 16y = 14\r<BR>\n" );
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document.write( "14x + 16y = -8\r<BR>\n" );
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document.write( "20x + 35y = 19\r<BR>\n" );
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document.write( "20x + 35y = 4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854432 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5360)</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=854432\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I see no blanks. I only see 6 linear equations on 2 variables.<BR>\n" );
document.write( "It is not a properly defined question, but I will answer it for educational purposes, making some assumptions.<BR>\n" );
document.write( "I am assuming that the answer expected is a system of 2 equations made using the equations listed.<BR>\n" );
document.write( "We can make 15 different systems of 2 equations with the equations listed.<BR>\n" );
document.write( "A system of 2 equations on 2 variables could have either a unique solution, or no solution, or infinite solutions.<BR>\n" );
document.write( "In this case, I will show below that we cannot form a system of equations with infinite solutions from the equation listed.<BR>\n" );
document.write( "The only systems of 2 equations with no unique solution that we can form the equations listed are systems with no solution.<BR>\n" );
document.write( "From the equations listed, we could form a system of 3 equations with no solution by taking any 3 of the equations listed, <BR>\n" );
document.write( "but I do not think that is the answer expected.<BR>\n" );
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document.write( "The graph of a linear equation on the x-y coordinate plane is a straight line.<BR>\n" );
document.write( "Two lines on a plane could<BR>\n" );
document.write( "be parallel, having no points in common, or<BR>\n" );
document.write( "intersect at only one point.<BR>\n" );
document.write( "If the graphs of two linear equations have 2 points in common,<BR>\n" );
document.write( "then both equations represent the same line, such as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B3y=5 ALIGN=MIDDLE  ALT=\"2x%2B3y=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%2B6y=10 ALIGN=MIDDLE  ALT=\"4x%2B6y=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "A system of two linear equations whose graph shows two intersecting straight line has one solution:<BR>\n" );
document.write( "the coordinates of the intersection point, the only ordered pair (x,y) that satisfies both equations.\r<BR>\n" );
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document.write( "On a coordinate plane, each of the equations listed would represent a straight line, as shown below.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C300%2C-4%2C4%2C-3%2C3%2C%0D%0A%2826-8x%29%2F12%2C%281-8x%29%2F12%2C%0D%0A%2814-14x%29%2F16%2C%28-8-14x%29%2F16%2C%0D%0A%2819-20x%29%2F35%2C%284-20x%29%2F35%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "There are 6 lines visible in the graph, and some visible intersection points,<BR>\n" );
document.write( "representing unique solutions for some of the possible systems of 2 linear equations.<BR>\n" );
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document.write( "SYSTEMS WITH UNIQUE SOLUTIONS:<BR>\n" );
document.write( "The graph for a system of 2 equations on 2 variables with a unique solution shows 2 lines intersecting at just one point.<BR>\n" );
document.write( "The coordinates of the intersection point are the unique solution.<BR>\n" );
document.write( "Those systems are not the answer we are looking for.<BR>\n" );
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document.write( "SYSTEMS WITH INFINITE SOLUTIONS:<BR>\n" );
document.write( "A system with infinite solutions does not have a unique solution, and could be a valid answer.<BR>\n" );
document.write( "The graph for a system of 2 equations on 2 variables with infinite solutions would show one line because the two equations represent the same line.<BR>\n" );
document.write( "That would be the case when you can obtain one of the equations by multiplying the other one by some number other than zero, as in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282x%2B3y=5%2C4x%2B6y=10%29 ALIGN=MIDDLE  ALT=\"system%282x%2B3y=5%2C4x%2B6y=10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "A system like that cannot be made from the equations listed.<BR>\n" );
document.write( "The only way to turn the left hand side of one equations into the left hand side of another is using the first 2,<BR>\n" );
document.write( "or the third and fourth equations, or fifth and fourth equations, and multiply by 1,<BR>\n" );
document.write( "but we would get the same equation and we see no repeats in the list of equations. <BR>\n" );
document.write( "We cannot do it with equations that have different left hand sides, because the ratios of the x and y coefficients are different.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2F12=3%2F4 ALIGN=MIDDLE  ALT=\"8%2F12=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the first two,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14%2F16=7%2F8 ALIGN=MIDDLE  ALT=\"14%2F16=7%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the third and fourth equations, and <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20%2F35=4%2F7 ALIGN=MIDDLE  ALT=\"20%2F35=4%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" > for third and fourth equations.<BR>\n" );
document.write( "We cannot turn one of the left hand sides <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%2B12y ALIGN=MIDDLE  ALT=\"8x%2B12y\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14x%2B+16y ALIGN=MIDDLE  ALT=\"14x%2B+16y\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20x%2B35y ALIGN=MIDDLE  ALT=\"20x%2B35y\"   DISABLED_style= \"max-width:90%; height:auto;\" > into another by multiplying by any number except zero. <BR>\n" );
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document.write( "So a systems with no unique solution that we can form from the list of \r<BR>\n" );
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document.write( "SYTEMS WITH NO SOLUTION:<BR>\n" );
document.write( "The graph for a  system of 2 equations on 2 variables with no solution shows 2 parallel lines.<BR>\n" );
document.write( "There is no point on the x-y coordinate plane that satisfies both equations.<BR>\n" );
document.write( "That is obvious for the system <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28system%288x%2B12y=26%2C8x%2B12y=1%29%29 ALIGN=MIDDLE  ALT=\"highlight%28system%288x%2B12y=26%2C8x%2B12y=1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > formed by the first 2 equations,<BR>\n" );
document.write( "because the value of the linear function on 2 variables <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%2Cy%29=8x%2B12y ALIGN=MIDDLE  ALT=\"f%28x%2Cy%29=8x%2B12y\"   DISABLED_style= \"max-width:90%; height:auto;\" > cannot be 26 and 1 for the same ordered pair (x,y).<BR>\n" );
document.write( "The slope of the lines represented by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%2B12y=26 ALIGN=MIDDLE  ALT=\"8x%2B12y=26\"   DISABLED_style= \"max-width:90%; height:auto;\" ><-><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-4%2F3%29%2B13%2F4 ALIGN=MIDDLE  ALT=\"y=%28-4%2F3%29%2B13%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%2B12y=1 ALIGN=MIDDLE  ALT=\"8x%2B12y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><-><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-4%2F3%29%2B1%2F4+is+%7B%7B%7B-3%2F4 ALIGN=MIDDLE  ALT=\"y=%28-4%2F3%29%2B1%2F4+is+%7B%7B%7B-3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Another system with no solution can be formed by the third and fourth equations; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28system%2814x+%2B+16y+=+14%2C%0D%0A14x+%2B+16y+=+-8%29%29 ALIGN=MIDDLE  ALT=\"highlight%28system%2814x+%2B+16y+=+14%2C%0D%0A14x+%2B+16y+=+-8%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "The same can be said of the system formed by the fifth and sixth equations:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28system%28%0D%0A20x+%2B+35y+=+19%2C%0D%0A20x+%2B+35y+=+4%29%29 ALIGN=MIDDLE  ALT=\"highlight%28system%28%0D%0A20x+%2B+35y+=+19%2C%0D%0A20x+%2B+35y+=+4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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