document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=123893>Question 123893</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this equation.  \r<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 #90888 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=90888\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Equations in this form [fractions on both sides] are called proportions. You can convert a proportion<BR>\n" );
document.write( "to a more conventional equation form by multiplying the numerator on one side by the denominator<BR>\n" );
document.write( "on the other side. If you to that for both numerators, you get two products and you separate the<BR>\n" );
document.write( "two products by an equal sign. Then just solve the resulting equation using algebraic<BR>\n" );
document.write( "procedures. Let's apply this method to your problem.<BR>\n" );
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document.write( "Begin by multiplying the numerator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2t+%2B+6 ALIGN=MIDDLE  ALT=\"2t+%2B+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the other side. The <BR>\n" );
document.write( "resulting distributed multiplication results in the product <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4t+%2B+12 ALIGN=MIDDLE  ALT=\"4t+%2B+12\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "Next multiply the numerator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the right side by the denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the left side. The <BR>\n" );
document.write( "resulting product is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5E2 ALIGN=MIDDLE  ALT=\"t%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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document.write( "Separate these two products with an equal sign and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5E2+=+4t+%2B+12 ALIGN=MIDDLE  ALT=\"t%5E2+=+4t+%2B+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get this in conventional quadratic form by subtracting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4t+%2B+12 ALIGN=MIDDLE  ALT=\"4t+%2B+12\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5E2+-+4t+-+12+=+0 ALIGN=MIDDLE  ALT=\"t%5E2+-+4t+-+12+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Factor the left side of this equation to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28t-6%29%2A%28t%2B4%29+=+0 ALIGN=MIDDLE  ALT=\"%28t-6%29%2A%28t%2B4%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice that this equation will be true if either of the factors on the left side equal zero<BR>\n" );
document.write( "because multiplying by a zero on the left side will make the entire left side equal to <BR>\n" );
document.write( "zero and therefore equal to the right side. To find the values of t that will make a factor<BR>\n" );
document.write( "equal to zero, just set the two factors equal to zero (one factor at a time) and solve for t.<BR>\n" );
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document.write( "First:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+-+6+=+0 ALIGN=MIDDLE  ALT=\"t+-+6+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Add 6 to both sides and this becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+6 ALIGN=MIDDLE  ALT=\"t+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Next:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+%2B+2+=+0 ALIGN=MIDDLE  ALT=\"t+%2B+2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Subtract 2 from both sides and this becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+t+=+-2 ALIGN=MIDDLE  ALT=\"+t+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So the two answers to this problem are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+6 ALIGN=MIDDLE  ALT=\"t+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+-2 ALIGN=MIDDLE  ALT=\"t+=+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can check these answers by substituting them one at a time into the original equation you<BR>\n" );
document.write( "were given and then working out the answers to make sure the left side of the equation does<BR>\n" );
document.write( "in fact equal the right side. For example, let's check to see if the given equation is<BR>\n" );
document.write( "equal on both sides when t is equal to +6. The original equation is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2Ft%29=%28%28t%29%2F%282t%2B6%29%29 ALIGN=MIDDLE  ALT=\"%282%2Ft%29=%28%28t%29%2F%282t%2B6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Replace t with +6 and this equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2F6%29=%28%286%29%2F%282%2A6%2B6%29%29 ALIGN=MIDDLE  ALT=\"%282%2F6%29=%28%286%29%2F%282%2A6%2B6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The denominator on the right side multiplies out to give 12 + 6 = 18 and this makes the<BR>\n" );
document.write( "equation reduce to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F6+=+6%2F18 ALIGN=MIDDLE  ALT=\"2%2F6+=+6%2F18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice that both sides reduce to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > so the equation is true when t = 6. <BR>\n" );
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document.write( "If you try the same check by substituting -2 for t you will find that is also works out such<BR>\n" );
document.write( "that both sides are equal. So both answers check.<BR>\n" );
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document.write( "Hope this helps you to understand the problem and how you can start to work out proportions by<BR>\n" );
document.write( "using cross multiplication to get them into a more conventional form.<BR>\n" );
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