document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1126889>Question 1126889</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find values for h and k so that the system has infinitely many solutions. Write answers as integers or fractions in lowest terms. \r<BR>\n" );
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document.write( "&#8722;8x &#8722; 6y = h<BR>\n" );
document.write( "12x + ky = 24\r<BR>\n" );
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document.write( "h= <BR>\n" );
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document.write( "k= <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 #743212 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</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=743212\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "both lines should have the same slope and y&#8722;intercept<BR>\n" );
document.write( " they are one line, coincident, meaning they have all points in common<BR>\n" );
document.write( " this means that there are an infinite number of solutions to the system\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8x+-6y+=+h ALIGN=MIDDLE  ALT=\"-8x+-6y+=+h\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12x+%2B+ky+=+24 ALIGN=MIDDLE  ALT=\"12x+%2B+ky+=+24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "first write both equations in slope-intercept form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8x+-6y+=+h ALIGN=MIDDLE  ALT=\"-8x+-6y+=+h\"   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=-8x+-h+=+6y ALIGN=MIDDLE  ALT=\"-8x+-h+=+6y\"   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=y=-%288%2F6%29x+-h%2F6+ ALIGN=MIDDLE  ALT=\"y=-%288%2F6%29x+-h%2F6+\"   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=y=-%284%2F3%29x+-h%2F6 ALIGN=MIDDLE  ALT=\"y=-%284%2F3%29x+-h%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > ............eq.1=> slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%284%2F3%29 ALIGN=MIDDLE  ALT=\"-%284%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and y-intercept is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-h%2F6 ALIGN=MIDDLE  ALT=\"-h%2F6\"   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=12x+%2B+ky+=+24 ALIGN=MIDDLE  ALT=\"12x+%2B+ky+=+24\"   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=ky+=+24-12x+ ALIGN=MIDDLE  ALT=\"ky+=+24-12x+\"   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=y+=+-%2812%2Fk%29x+%2B24%2Fk ALIGN=MIDDLE  ALT=\"y+=+-%2812%2Fk%29x+%2B24%2Fk\"   DISABLED_style= \"max-width:90%; height:auto;\" >.............eq.2 => slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%2812%2Fk%29 ALIGN=MIDDLE  ALT=\"-%2812%2Fk%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and y-intercept is<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+24%2Fk ALIGN=MIDDLE  ALT=\"+24%2Fk\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "equal slopes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%284%2F3%29=-%2812%2Fk%29 ALIGN=MIDDLE  ALT=\"-%284%2F3%29=-%2812%2Fk%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" >; both sides multiply by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   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=4%2F3=12%2Fk ALIGN=MIDDLE  ALT=\"4%2F3=12%2Fk\"   DISABLED_style= \"max-width:90%; height:auto;\" >........cross multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4k=12%2A3 ALIGN=MIDDLE  ALT=\"4k=12%2A3\"   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=4k=36 ALIGN=MIDDLE  ALT=\"4k=36\"   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=k=9 ALIGN=MIDDLE  ALT=\"k=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "equal y-intercepts:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-h%2F6=24%2Fk+ ALIGN=MIDDLE  ALT=\"-h%2F6=24%2Fk+\"   DISABLED_style= \"max-width:90%; height:auto;\" >........substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   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=-h%2F6=24%2F9 ALIGN=MIDDLE  ALT=\"-h%2F6=24%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > ......cross multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9h=24%2A6 ALIGN=MIDDLE  ALT=\"-9h=24%2A6\"   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=h=144%2F-9 ALIGN=MIDDLE  ALT=\"h=144%2F-9\"   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=h=-16 ALIGN=MIDDLE  ALT=\"h=-16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, your lines are:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%284%2F3%29x+-%28-16%29%2F6 ALIGN=MIDDLE  ALT=\"y=-%284%2F3%29x+-%28-16%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > -><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%284%2F3%29x+%2B8%2F3 ALIGN=MIDDLE  ALT=\"y=-%284%2F3%29x+%2B8%2F3\"   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=y+=+-%2812%2F9%29x+%2B24%2F9 ALIGN=MIDDLE  ALT=\"y+=+-%2812%2F9%29x+%2B24%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" >-><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-%284%2F3%29x+%2B8%2F3 ALIGN=MIDDLE  ALT=\"y+=+-%284%2F3%29x+%2B8%2F3\"   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=+graph%28600%2C+600%2C+-10%2C10%2C+-10%2C+10%2C+-%284%2F3%29x+%2B8%2F3%2C+-%284%2F3%29x+%2B8%2F3%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "</TD></TR></TABLE>\n" );