document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=797005>Question 797005</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have two Questions. My first one is, How do you algebraically figure out if vales of x and y can be the same?\r<BR>\n" );
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document.write( "For example, In this equation 12= 2x + 3y I can figure out through guess and check that the number 2.4 can be used for x and y at the same time.\r<BR>\n" );
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document.write( "Now for my next question.\r<BR>\n" );
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document.write( "Say you have something like this:\r<BR>\n" );
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document.write( "24= 4x + 8y\r<BR>\n" );
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document.write( "How do I determine algebraically what values of y and x will make 24 equal to 0?\r<BR>\n" );
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document.write( "Thanks so much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #481662 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=481662\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12=+2x+%2B+3y ALIGN=MIDDLE  ALT=\"12=+2x+%2B+3y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=y ALIGN=MIDDLE  ALT=\"x=y\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then I can substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > 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;\" > and re-write the equation as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12=+2x+%2B+3x ALIGN=MIDDLE  ALT=\"12=+2x+%2B+3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12=5x ALIGN=MIDDLE  ALT=\"12=5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2F5=x ALIGN=MIDDLE  ALT=\"12%2F5=x\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2.4 ALIGN=MIDDLE  ALT=\"x=2.4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24=+4x+%2B+8y ALIGN=MIDDLE  ALT=\"24=+4x+%2B+8y\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > are related by that equation, and we can find some of the infinite pairs of values (x,y) that satisfy that equation.<BR>\n" );
document.write( "(Finding infinite pairs would take infinite time, and I only have a few minutes until my bedtime).<BR>\n" );
document.write( "24 will never be 0, or anything other than 24.<BR>\n" );
document.write( "However, maybe you are thinking of finding what value of x corresponds to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for that equation, and what value of y corresponds to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > for that equation.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24=+4%2A0%2B8y ALIGN=MIDDLE  ALT=\"24=+4%2A0%2B8y\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24=0%2B8y ALIGN=MIDDLE  ALT=\"24=0%2B8y\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24=8y ALIGN=MIDDLE  ALT=\"24=8y\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24%2F8=y ALIGN=MIDDLE  ALT=\"24%2F8=y\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3 ALIGN=MIDDLE  ALT=\"y=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the pair (0,3), with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3 ALIGN=MIDDLE  ALT=\"y=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > satisfies the relation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24=+4x+%2B+8y ALIGN=MIDDLE  ALT=\"24=+4x+%2B+8y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24=4x%2B8%2A0 ALIGN=MIDDLE  ALT=\"24=4x%2B8%2A0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24=4x%2B0 ALIGN=MIDDLE  ALT=\"24=4x%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24=4x ALIGN=MIDDLE  ALT=\"24=4x\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24%2F4=x ALIGN=MIDDLE  ALT=\"24%2F4=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the pair (6,0), with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > satisfies the relation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24=+4x+%2B+8y ALIGN=MIDDLE  ALT=\"24=+4x+%2B+8y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Those two data pairs, and the infinite pairs (x,y) that satisfy <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24=+4x+%2B+8y ALIGN=MIDDLE  ALT=\"24=+4x+%2B+8y\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be graphed as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-2%2C8%2C-3%2C7%2C%0D%0Agrid%281%29%2C%0D%0Ablue%28circle%286%2C0%2C0.2%29%29%2Cblue%28circle%280%2C3%2C0.2%29%29%2C%0D%0Ablue%28line%28-8%2C7%2C12%2C-3%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Equations like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12=+2x+%2B+3y ALIGN=MIDDLE  ALT=\"12=+2x+%2B+3y\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24=+4x+%2B+8y ALIGN=MIDDLE  ALT=\"24=+4x+%2B+8y\"   DISABLED_style= \"max-width:90%; height:auto;\" > are called linear equations and their graphs are straight line.<BR>\n" );
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document.write( "If an equation has an x, and/or a y, and the x and/or y are at most multiplied times numbers and added or subtracted (no exponents, no x or y in a denominator, no other fancy stuff), it is a linear equation.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28x=-2%29 ALIGN=MIDDLE  ALT=\"blue%28x=-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28y=sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"red%28y=sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28y=2x%2B1%29 ALIGN=MIDDLE  ALT=\"green%28y=2x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are linear equations <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-5%2C5%2C-5%2C5%2C%0D%0Agrid%280%29%2C%0D%0Ablue%28line%28-2%2C-5%2C-2%2C5%29%29%2C%0D%0Ared%28line%28-5%2Csqrt%283%29%2C5%2Csqrt%283%29%29%29%2C%0D%0Agreen%28line%28-5%2C-9%2C5%2C11%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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